[User]
Let ACDE be a square with center B and circumradius 1. Line K is the perpendicular bisector of segment EC. Point U is the midpoint of segment CD. Let H be the line through point D perpendicular to line K. Let P be the reflection of point A across point E. Point J is the reflection of point U across line H. Find the length of segment JP. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center P and circumradius 3. Line Z passes through points H and C. Let S be the midpoint of segment BG. Point Y is the reflection of point S across line Z. Point U is the midpoint of segment YE. What is the distance between points U and Y? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 7. Let S be the perpendicular bisector of segment EI. Line T passes through points L and I. Let W be the reflection of line T across line S. Lines W and T intersect at point N. Find the measure of angle HDN, in radians. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJ be a regular 10-gon with center W and circumradius 7. Let K be the line through points B and E. Point S is the reflection of point A across line K. Point N is the midpoint of segment SW. Point T is the midpoint of segment DN. What is the distance between points H and T? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center Z and circumradius 5. Point R is the midpoint of segment AZ. Let O be the circumcircle of polygon ABCDE. Point W is the midpoint of segment DR. Point S is the midpoint of segment DW. Circle L is the reflection of circle O across point S. Let X be the inversion of point B in circle L. Let J be the midpoint of segment ZX. What is the length of segment JA? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center X and circumradius 8. Point P is obtained by rotating point X by angle pi/2 about point D. Point M is the midpoint of segment XE. Point Y is the reflection of point P across point M. Point T is the reflection of point D across segment MY. What is the length of segment TP? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDFGHIJKLM be a regular dodecagon with center E and circumradius 1. Line Q is the perpendicular bisector of segment BI. Let X be the reflection of line Q through point J. Let P be the reflection of point E across point D. Let W be the reflection of point P across point J. Point Z is the reflection of point L across line X. Point T is the reflection of point W across point Z. What is the distance between points T and F? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center Q and circumradius 3. Let Z be the midpoint of segment DC. Let T be the perpendicular bisector of segment ZE. Point J is on line T. Point G is the reflection of point D across segment JE. What is the length of segment GE? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
MI = 3. ABCDEFGHJKLN is a regular dodecagon with center I and circumradius 1. Line O is the angle bisector of angle BCH. Line P passes through points E and J. Line V is the reflection of line P across line O. Point S is the reflection of point L across line V. Point Y is the midpoint of segment SG. What is the length of segment YF? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEGHIJ is a regular 9-gon with center F and circumradius 3. Let Z be the perpendicular bisector of segment AG. Point Q lies on line Z at distance 10 from point G. Let O be the reflection of line Z through point H. Line W is the reflection of line O across point B. Let R be the reflection of line W through point G. Point U is the reflection of point Q across line R. What is the distance between points A and U? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDF is a regular 5-gon with center E and circumradius 7. Line H passes through points B and D. Let V be the reflection of line H through point F. Point L is the reflection of point A across line H. Point R is the reflection of point L across point D. Let J be the reflection of point F across point R. Point Q is the reflection of point J across line V. Point I is the reflection of point Q across segment RA. Find the length of segment IQ. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDFG be a regular hexagon with center E and circumradius 8. Let M be the reflection of point A across point C. Point Y is the reflection of point G across segment MA. Let W be the reflection of point M across point A. Find the distance between points Y and W. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCD be a square with center Y and circumradius 10. Point T is the midpoint of segment BY. Point E is the midpoint of segment CD. Let N be the perpendicular bisector of segment EY. Line O passes through point T perpendicular to line N. Point Z is the reflection of point A across line O. Let U be the midpoint of segment AZ. Let H be the midpoint of segment DE. Find the distance between points H and U. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center N and circumradius 6. Let Z be the line through points D and C. Let M be the reflection of line Z through point F. Point X is the reflection of point F across line M. Let W be the midpoint of segment DN. What is the length of segment WX? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJ is a regular 10-gon with center L and circumradius 12. Point T is the midpoint of segment AI. Point X is the reflection of point C across segment TH. What is the distance between points D and X? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center Y and circumradius 2. Let U be the rotation of point C by angle pi around point D. Point Z is the reflection of point A across point D. Let W be the perpendicular bisector of segment ZY. Line X is the reflection of line W across point C. Line Q is the reflection of line W across line X. Point J is the reflection of point D across line Q. Find the distance between points U and J. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center W and circumradius 9. Point X is the reflection of point A across point B. Line N passes through points A and W. Let Y be the line through points X and E. Line S is the reflection of line N across line Y. Point M is the intersection of lines S and N. Let I be the midpoint of segment DM. Point H is the midpoint of segment DI. Find the length of segment FH. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDE is a regular 5-gon with center Q and circumradius 11. Let P be the reflection of point A across point E. Let M be the circumcircle of polygon ABCDE. Let U be the angle bisector of angle QPD. Let K be the reflection of circle M across line U. Let W be the inversion of point A in circle K. Find the distance between points C and W. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center V and circumradius 1. Line O passes through points B and E. Let H be the angle bisector of angle CDA. Let J be the reflection of line H across line O. Line T passes through point C perpendicular to line J. Point U is the reflection of point D across line T. Point Z is the midpoint of segment UA. What is the distance between points A and Z? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJ is a regular 10-gon with center Q and circumradius 6. Line N is the perpendicular bisector of segment HF. Let O be the reflection of line N through point H. Point Z is the reflection of point A across line O. What is the length of segment ZH? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center V and circumradius 7. Let O be the circumcircle of polygon ABCDE. Circle N is the reflection of circle O across point D. Let P be the line through points V and D. Let Y be the inversion of point E in circle N. Let S be the perpendicular bisector of segment DY. Line W passes through point C parallel to line S. Point G is the reflection of point C across point E. Lines P and W intersect at point Q. Find the distance between points G and Q. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFG is a regular 7-gon with center I and circumradius 9. Point M is the midpoint of segment GI. Let S be the circumcircle of polygon ABCDEFG. Let T be the angle bisector of angle IGE. Circle W is the reflection of circle S across line T. Let P be the reflection of circle W through point E. Point Z is the inversion of point M with respect to circle P. What is the length of segment AZ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center K and circumradius 9. Line J passes through points A and C. Line U is the angle bisector of angle ACD. Line W passes through point F parallel to line U. Lines W and J intersect at point T. What is the measure of angle CTB, in radians? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center M and circumradius 9. Line R is the perpendicular bisector of segment DC. Let Y be the reflection of point E across point B. Let Z be the line through point C perpendicular to line R. Point V is the midpoint of segment YC. Point T is the reflection of point A across line Z. What is the distance between points V and T? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFG be a regular 7-gon with center Z and circumradius 10. Line M passes through points C and A. Let Q be the line through point D perpendicular to segment GA. Point I is the reflection of point Z across line M. Line X is the reflection of line M across line Q. Line H passes through points Z and I. Line N is the reflection of line H across line X. Let J be the reflection of line Q across line N. Lines M and J intersect at point O. Find the distance between points O and Z. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDE is a regular 5-gon with center N and circumradius 10. Let J be the midpoint of segment ND. Let R be the perpendicular bisector of segment NB. Point I is the reflection of point J across line R. Point V is the midpoint of segment BI. What is the length of segment IV? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABDEFGHIJ be a regular 9-gon with center C and circumradius 9. Point Z is the reflection of point F across point C. Line T passes through points E and A. Point Q is the reflection of point Z across line T. Let Y be the line through points A and Z. Point K is the reflection of point B across line Y. What is the distance between points Q and K? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 5. Let O be the reflection of point F across point J. Let M be the midpoint of segment IK. Let W be the midpoint of segment MG. What is the distance between points W and O? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJK be a regular 11-gon with center V and circumradius 10. Let L be the line through points A and C. Point O is the reflection of point E across line L. Line W passes through points D and C. Point R is the reflection of point F across line W. What is the distance between points R and O? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHJ is a regular 9-gon with center I and circumradius 2. Let N be the rotation of point F by angle 2pi/3 around point B. Point L is the reflection of point N across point E. Let U be the midpoint of segment JI. Point Y is the midpoint of segment AU. What is the distance between points L and Y? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 4. Let Z be the circumcircle of polygon ABCDEFGHIJKL. Let T be the reflection of point A across point I. Circle X is the reflection of circle Z across point L. Point Y is the inversion of point B with respect to circle X. What is the distance between points Y and T? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center Y and circumradius 7. Point J is the midpoint of segment AF. Point U is obtained by rotating point C by angle 7pi/12 about point J. Line Z passes through points U and C. Line G passes through point B parallel to line Z. Point M is the intersection of line Z and segment EJ. Point T is the reflection of point J across line G. Point S is the reflection of point M across segment CJ. Find the length of segment TS. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHI be a regular 9-gon with center P and circumradius 7. Point J is the reflection of point D across point G. Let V be the line through point I perpendicular to segment CB. Let N be the reflection of line V through point J. Point U is the reflection of point G across line N. Find the distance between points E and U. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJ be a regular 10-gon with center W and circumradius 8. Point P is the midpoint of segment JH. Let V be the reflection of point C across point P. Point K is the midpoint of segment JB. What is the distance between points V and K? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFG be a regular 7-gon with center I and circumradius 7. Let L be the circumcircle of polygon ABCDEFG. Circle N is the reflection of circle L across point F. Point Y is the inversion of point A with respect to circle N. Let V be the perpendicular bisector of segment IC. Let K be the line through point C perpendicular to line V. Point R is the reflection of point Y across line K. Let O be the midpoint of segment RB. Let X be the midpoint of segment DB. Point S is the midpoint of segment XA. Find the length of segment OS. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJ is a regular 10-gon with center N and circumradius 6. Point Z is the midpoint of segment DA. Point L is the midpoint of segment ZJ. Point O is the reflection of point D across point L. Find the distance between points I and O. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJK is a regular 11-gon with center U and circumradius 8. Let Y be the line through points H and K. Line Q passes through points C and B. Line V is the reflection of line Y across point J. Point S is the intersection of lines V and Q. What is the distance between points S and B? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center K and circumradius 9. Let I be the circumcircle of polygon ABCDEF. Let N be the midpoint of segment BF. Point P is the reflection of point N across point E. Point Z is the inversion of point P with respect to circle I. Find the length of segment FZ. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center O and circumradius 1. Point J is the midpoint of segment FE. Point W is the reflection of point C across segment DB. Point S is the midpoint of segment WF. Point Y is the reflection of point J across point O. What is the distance between points S and Y? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center T and circumradius 2. Let N be the line through points T and G. Line Y is the perpendicular bisector of segment TF. Line U is the reflection of line Y across line N. Point K is the reflection of point A across line U. Find the length of segment KC. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center O and circumradius 5. Line X passes through points B and F. Let G be the perpendicular bisector of segment OD. Point M is the reflection of point D across line X. Let R be the angle bisector of angle ABM. Line J is the reflection of line G across line R. Lines X and J intersect at point Q. What is the length of segment QM? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJ be a regular 10-gon with center P and circumradius 12. Let Y be the perpendicular bisector of segment GB. Line L is the reflection of line Y across point B. Line R passes through point G perpendicular to line L. Point Z is on line R. Point N is the reflection of point Z across point B. Find the measure of angle EPN, in radians. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHI be a regular 9-gon with center U and circumradius 12. Point Y is obtained by rotating point I by angle pi/2 about point A. Point V is the midpoint of segment EI. Let K be the line through point V parallel to segment IA. Line J passes through point H perpendicular to line K. Point R is the reflection of point Y across line J. Find the length of segment RF. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 10. Line X is the angle bisector of angle AHJ. Point M is the reflection of point G across line X. Point Z is the midpoint of segment CB. Point P is the reflection of point M across segment LC. What is the distance between points Z and P? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center N and circumradius 3. Point T is obtained by rotating point C by angle 2pi/3 about point A. Let W be the line through point H parallel to segment NF. Let Y be the line through point T perpendicular to line W. Point S is the reflection of point A across line Y. Find the distance between points S and N. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJKL is a regular dodecagon with center M and circumradius 1. Line O is the perpendicular bisector of segment GD. Point W is the midpoint of segment FM. Let P be the midpoint of segment WG. Let Q be the line through point P perpendicular to line O. Point X is the reflection of point B across line Q. What is the length of segment XP? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJ is a regular 10-gon with center L and circumradius 5. Let R be the circumcircle of polygon ABCDEFGHIJ. Line Z is the perpendicular bisector of segment CL. Point Q is the midpoint of segment BD. Let U be the reflection of circle R across line Z. Let W be the inversion of point Q in circle U. Find the distance between points C and W. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 1. Point V is the reflection of point A across point L. Point Q is the reflection of point C across point G. Let N be the angle bisector of angle ZKJ. Line O passes through point H parallel to line N. Let R be the reflection of line O across line N. Point W is the reflection of point Q across line R. What is the distance between points V and W? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center M and circumradius 12. Line H passes through point A perpendicular to segment DM. Let J be the line through point D parallel to line H. Point Y is the reflection of point M across line J. Let L be the reflection of point M across point Y. What is the distance between points F and L? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCD be a square with center K and circumradius 4. Line W is the angle bisector of angle BAC. Let M be the midpoint of segment KD. Line U is the reflection of line W across point B. Point Z is the reflection of point D across line U. Point O is the reflection of point D across point Z. Point P is the reflection of point O across segment DM. What is the length of segment PA? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEGHIJK is a regular 10-gon with center F and circumradius 6. Line S is the angle bisector of angle IGB. Let O be the reflection of line S through point J. Let V be the reflection of line O through point E. Line X is the reflection of line V across point H. Point M is the reflection of point D across line X. Find the distance between points M and H. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center H and circumradius 1. Let L be the reflection of point E across point A. Point Q is the midpoint of segment LC. Let U be the reflection of point L across point E. Point X is the midpoint of segment QL. What is the distance between points X and U? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCEFGHI be a regular octagon with center D and circumradius 4. Point V is the midpoint of segment EA. Line J is the angle bisector of angle FGV. Let N be the line through point F parallel to line J. Line X is the reflection of line J across line N. Point P is the midpoint of segment BF. Point O is the reflection of point P across line X. Find the distance between points A and O. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center L and circumradius 7. Let J be the reflection of point A across point E. Let M be the circumcircle of polygon ABCDEFGH. Circle K is the reflection of circle M across point G. Let X be the inversion of point B in circle K. What is the length of segment JX? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHJK is a regular 10-gon with center I and circumradius 8. Let R be the angle bisector of angle GKC. Let Q be the line through point F parallel to line R. Let M be the reflection of line Q across line R. Point Y is the reflection of point C across line M. Find the distance between points J and Y. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHJKLM be a regular dodecagon with center I and circumradius 1. Let Z be the line through points G and F. Line Q passes through point I parallel to line Z. Line W is the reflection of line Z across line Q. Point N is the reflection of point G across line W. Find the distance between points H and N. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABDEFGHIJKLM is a regular dodecagon with center C and circumradius 11. Let N be the reflection of point D across point K. Point P is the midpoint of segment NG. Let T be the midpoint of segment LJ. Find the distance between points P and T. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDFGHIJK be a regular 10-gon with center E and circumradius 8. Line S passes through points H and G. Line Y passes through point J parallel to line S. Point R is the reflection of point H across point A. Let X be the midpoint of segment GR. Line Y and segment HK intersect at point L. Find the length of segment XL. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center Y and circumradius 11. Line Z is the angle bisector of angle FEA. Point S is the reflection of point D across line Z. Point U is the reflection of point G across point S. Point L is the midpoint of segment HC. Find the length of segment LU. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center L and circumradius 1. Let Q be the circumcircle of polygon ABCDEF. Circle X is the reflection of circle Q across point E. Point W is the inversion of point F with respect to circle X. Point O is the inversion of point W with respect to circle Q. What is the length of segment OW? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center G and circumradius 3. Let O be the midpoint of segment BD. Point Z is the reflection of point A across segment DB. Point H is the reflection of point C across point Z. Find the length of segment HO. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJKL is a regular dodecagon with center U and circumradius 6. Let Q be the reflection of point K across point A. Let X be the midpoint of segment UB. Let Z be the midpoint of segment FQ. Find the length of segment ZX. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJ be a regular 10-gon with center T and circumradius 9. Point Z is the reflection of point G across point B. Point L is the midpoint of segment FZ. Point K is the midpoint of segment DZ. Find the distance between points K and L. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDFGHIJK is a regular 10-gon with center E and circumradius 9. Let X be the line through points B and J. Point N is the intersection of segment AD and line X. Point P is the reflection of point J across point A. Let S be the midpoint of segment NB. Find the length of segment PS. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center U and circumradius 2. Let T be the reflection of point G across point B. Let S be the midpoint of segment GB. Point R is the reflection of point T across point S. Find the length of segment RA. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDE is a regular 5-gon with center S and circumradius 12. Let I be the perpendicular bisector of segment BS. Point P is the reflection of point E across line I. Line X passes through point D perpendicular to line I. Line L passes through point E perpendicular to line X. Point W is the reflection of point C across line L. What is the length of segment PW? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 5. Let Q be the circumcircle of polygon ABDEFGHIJKLM. Point T is the reflection of point B across point K. Line V passes through points C and F. Let Y be the line through point T perpendicular to line V. Point Z is the reflection of point G across line Y. Point O is the inversion of point Z with respect to circle Q. Find the distance between points A and O. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIK is a regular 10-gon with center J and circumradius 3. Let P be the circumcircle of polygon ABCDEFGHIK. Let N be the perpendicular bisector of segment JE. Circle R is the reflection of circle P across point H. Point Q is the reflection of point D across line N. Circle T is the reflection of circle R across line N. Let S be the inversion of point Q in circle T. What is the length of segment QS? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 6. Point Q is the midpoint of segment HJ. Let O be the midpoint of segment EC. Point T is the reflection of point B across point O. Find the distance between points Q and T. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center O and circumradius 2. Line I passes through point C parallel to segment OA. Point Z is the reflection of point D across line I. Point T is the midpoint of segment ZE. Find the distance between points O and T. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center X and circumradius 12. Let T be the circumcircle of polygon ABCDEFGHIJKL. Circle Y is the reflection of circle T across point E. Circle V is the reflection of circle T across point K. Point S is the inversion of point F with respect to circle V. Let Q be the inversion of point S in circle Y. Find the distance between points G and Q. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center Z and circumradius 1. Let T be the midpoint of segment CE. Let U be the angle bisector of angle TFB. Let S be the line through point C perpendicular to line U. Point R is the reflection of point F across line S. What is the length of segment BR? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center G and circumradius 11. Point L is the reflection of point A across point D. Let V be the perpendicular bisector of segment CF. Let M be the line through point L perpendicular to line V. Point X is the reflection of point D across line M. What is the distance between points G and X? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center N and circumradius 10. Point V is the midpoint of segment HF. Let R be the perpendicular bisector of segment AV. Point J is the reflection of point G across line R. Find the distance between points G and J. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center P and circumradius 8. Point R is the midpoint of segment AF. Let K be the perpendicular bisector of segment BR. Let Z be the reflection of line K through point A. Point H is the reflection of point R across point D. Point I is the reflection of point H across line Z. What is the distance between points I and E? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center G and circumradius 11. Point R is obtained by rotating point G by angle pi/4 about point B. Let T be the midpoint of segment ER. Let M be the circumcircle of polygon ABCDEF. Let O be the midpoint of segment FT. Point V is the reflection of point D across point B. Circle U is the reflection of circle M across point O. Let X be the inversion of point F in circle U. What is the length of segment VX? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center U and circumradius 1. Point Q is the midpoint of segment UE. Point G is the midpoint of segment QC. Point T is the reflection of point G across point C. Point X is the reflection of point T across segment AQ. Find the distance between points T and X. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center Y and circumradius 6. Let T be the perpendicular bisector of segment DB. Line Z is the angle bisector of angle HGA. Line L is the reflection of line Z across line T. Let W be the rotation of point Y by angle 11pi/12 around point A. Let R be the circumcircle of polygon ABCDEFGH. Circle S is the reflection of circle R across point W. Let O be the reflection of circle S across line L. Point U is the inversion of point G with respect to circle O. Find the length of segment UH. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJ be a regular 10-gon with center K and circumradius 2. Point N is obtained by rotating point B by angle 11pi/12 about point C. Let W be the perpendicular bisector of segment NE. Line Q passes through point I perpendicular to line W. Let M be the line through point C parallel to line Q. Line M and segment KD intersect at point Y. Find the distance between points Y and A. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABDEFG is a regular hexagon with center C and circumradius 8. Let K be the line through points A and D. Point X is the reflection of point F across point B. Let L be the reflection of point G across point D. Point U is the reflection of point X across segment EB. Point T is the reflection of point L across line K. What is the length of segment TU? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ACDEFGHIJKLM is a regular dodecagon with center B and circumradius 11. Line O passes through points A and F. Point S is the reflection of point C across point D. Line W is the perpendicular bisector of segment KS. Let Y be the reflection of line W across line O. Point Z is the intersection of lines Y and O. What is the length of segment ZI? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFHI be a regular octagon with center G and circumradius 4. Line X passes through points I and G. Point M is the reflection of point I across point F. Point U is the reflection of point M across line X. Let T be the midpoint of segment UC. What is the length of segment TM? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center K and circumradius 4. Let Y be the midpoint of segment CK. Let W be the reflection of point B across point A. Point R is the reflection of point Y across segment DW. What is the length of segment CR? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCEFGHI is a regular octagon with center D and circumradius 12. Line X is the angle bisector of angle FBI. Line M is the reflection of line X across point H. Point W is the reflection of point D across line M. Point Y is the midpoint of segment WG. Find the length of segment YC. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJ is a regular 10-gon with center S and circumradius 8. Line O passes through point D perpendicular to segment DJ. Point Z is the reflection of point S across line O. Let K be the reflection of point I across point B. Find the length of segment KZ. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDFGHI is a regular octagon with center E and circumradius 7. Let J be the line through points I and C. Point T is the reflection of point B across line J. Let M be the midpoint of segment AT. Point R is the reflection of point D across point M. Find the distance between points R and A. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEGHIJ is a regular 9-gon with center F and circumradius 9. Let O be the reflection of point E across point H. Let W be the angle bisector of angle CBG. Point Q is the reflection of point F across line W. Let Y be the reflection of line W through point Q. Point T is the reflection of point F across line Y. What is the length of segment OT? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGIJ is a regular 9-gon with center H and circumradius 10. Let V be the line through points H and E. Line K is the perpendicular bisector of segment CH. Let N be the reflection of line K through point D. Point U is the reflection of point A across line N. Point Y is the intersection of lines V and N. What is the length of segment YU? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHI be a regular 9-gon with center U and circumradius 5. Point S is the midpoint of segment UD. Point O is the reflection of point I across point D. Line Q passes through point O perpendicular to segment ID. Line P is the reflection of line Q across point B. Line Y is the reflection of line P across line Q. Point X is the reflection of point A across line Y. Find the distance between points X and S. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center Y and circumradius 10. Point U is obtained by rotating point B by angle 3pi/4 about point F. Point K is the reflection of point U across point F. Point Z is the reflection of point U across point B. Line N is the perpendicular bisector of segment BK. Point W is the reflection of point A across line N. What is the length of segment ZW? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center F and circumradius 1. Let L be the angle bisector of angle ACF. Point W is the midpoint of segment ED. Point H is the reflection of point W across line L. Point X is the midpoint of segment DC. Point J is the midpoint of segment HX. Find the distance between points H and J. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center T and circumradius 3. Point U is the midpoint of segment BD. Point N is the reflection of point A across segment DU. Point Z is the midpoint of segment ND. Let R be the midpoint of segment ZE. Find the length of segment RN. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJ be a regular 10-gon with center K and circumradius 3. Let P be the line through points C and B. Line L is the angle bisector of angle KFG. Line Y is the reflection of line L across line P. Point V is the reflection of point B across line Y. What is the distance between points H and V? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center P and circumradius 12. Let X be the circumcircle of polygon ABCDEFGH. Point O is the midpoint of segment CB. Let K be the inversion of point O in circle X. Let Z be the midpoint of segment KF. What is the distance between points Z and E? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDE is a regular 5-gon with center M and circumradius 8. Let F be the angle bisector of angle CBM. Let S be the line through point M perpendicular to line F. Let Z be the reflection of line S through point C. Let Q be the line through point E parallel to line Z. Line N is the reflection of line F across point E. Line Z and segment AE intersect at point O. Lines Q and N intersect at point W. What is the length of segment WO? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ACDEF is a regular 5-gon with center B and circumradius 6. Let O be the circumcircle of polygon ACDEF. Let Y be the reflection of circle O through point D. Let K be the reflection of point F across point E. Let U be the inversion of point A in circle Y. Let R be the line through point D parallel to segment CU. Point V is the reflection of point K across line R. What is the length of segment VK? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center K and circumradius 3. Let P be the circumcircle of polygon ABCDEF. Point N is the reflection of point A across segment CE. Let R be the reflection of point C across point N. Point S is the inversion of point R with respect to circle P. Find the length of segment SD. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center Y and circumradius 2. Let Z be the reflection of point D across point Y. Point S is the reflection of point A across segment BZ. Let W be the midpoint of segment ZB. Find the distance between points W and S. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCD is a square with center H and circumradius 7. Let T be the midpoint of segment AB. Let E be the circumcircle of polygon ABCD. Let G be the inversion of point T in circle E. Point X is the reflection of point G across point H. Point Q is the reflection of point A across point X. Find the distance between points Q and X. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJ is a regular 10-gon with center L and circumradius 12. Let Q be the rotation of point G by angle pi/4 around point F. Point T is the midpoint of segment BJ. Let X be the circumcircle of polygon ABCDEFGHIJ. Let R be the midpoint of segment JF. Let N be the inversion of point T in circle X. Let Y be the line through point I perpendicular to segment RA. Let P be the line through point Q parallel to line Y. Point U is the reflection of point N across line P. What is the distance between points J and U? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEGHIJ be a regular 9-gon with center F and circumradius 5. Point L is the midpoint of segment FJ. Let P be the midpoint of segment LC. Point W is the midpoint of segment PI. Let U be the line through points W and A. Segment BH and line U intersect at point N. What is the distance between points N and W? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCD is a square with center N and circumradius 3. Let F be the rotation of point N by angle 7pi/12 around point B. Let M be the perpendicular bisector of segment NC. Let U be the perpendicular bisector of segment AC. Point Q is the reflection of point F across line M. Line L passes through point Q parallel to line U. Point K is the reflection of point F across line L. What is the distance between points K and Q? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center X and circumradius 3. Let W be the line through point H perpendicular to segment GE. Point T is the reflection of point C across line W. Let J be the midpoint of segment FT. Point R is the reflection of point J across segment GE. What is the distance between points R and G? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center I and circumradius 6. Let O be the angle bisector of angle ACD. Let U be the circumcircle of polygon ABCDEFGH. Circle L is the reflection of circle U across line O. Let T be the inversion of point B in circle L. Let S be the midpoint of segment BH. What is the distance between points S and T? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJKL is a regular dodecagon with center P and circumradius 2. Line O is the angle bisector of angle HJP. Point U is the reflection of point C across line O. Let Z be the midpoint of segment KU. Point R is the midpoint of segment BZ. What is the distance between points R and A? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEG be a regular hexagon with center F and circumradius 2. Let T be the line through points B and G. Let M be the reflection of point G across point D. Let W be the reflection of point M across point E. Point K is the reflection of point W across line T. What is the length of segment KA? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIK be a regular 10-gon with center J and circumradius 8. Line R is the perpendicular bisector of segment AH. Let Y be the line through point F parallel to line R. Point P is the midpoint of segment CG. Point X is the midpoint of segment DG. Point S is the reflection of point X across line Y. Find the distance between points P and S. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center N and circumradius 9. Line Y is the angle bisector of angle HND. Line O is the reflection of line Y across point G. Point L is the reflection of point E across line O. Point I is the intersection of segment LF and line O. Find the distance between points I and B. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center K and circumradius 9. Line U passes through points A and E. Let S be the reflection of point D across point E. Line Y passes through point F perpendicular to line U. Point R is the reflection of point S across line Y. Let P be the midpoint of segment SF. What is the length of segment PR? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFG be a regular 7-gon with center M and circumradius 9. Let P be the circumcircle of polygon ABCDEFG. Point J is the midpoint of segment BA. Let O be the inversion of point J in circle P. What is the distance between points O and J? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEGHIJKLM be a regular dodecagon with center F and circumradius 9. Point W is the reflection of point E across point J. Let O be the line through point I parallel to segment LM. Point S is the reflection of point W across line O. What is the length of segment MS? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDE is a regular 5-gon with center Q and circumradius 4. Point L is the reflection of point D across point A. Point Y is the reflection of point L across segment CE. Let Z be the midpoint of segment CE. What is the distance between points Z and Y? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 12. Let Z be the line through points M and A. Point X is the reflection of point I across point B. Let Q be the line through point F parallel to line Z. Point U is the reflection of point X across line Q. What is the length of segment UX? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCEFGHIJK be a regular 10-gon with center D and circumradius 8. Let S be the line through points A and J. Point T is the midpoint of segment CE. Let M be the line through point F perpendicular to line S. Point O is the reflection of point T across line M. What is the length of segment OI? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHJKL be a regular 11-gon with center I and circumradius 11. Let Q be the rotation of point A by angle 2pi/3 around point I. Point M is the reflection of point L across segment JD. Let Y be the midpoint of segment JQ. Find the length of segment MY. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDF is a regular 5-gon with center E and circumradius 12. Let Y be the midpoint of segment BD. Point Q is the reflection of point Y across point C. Let K be the circumcircle of polygon ABCDF. Point R is the inversion of point Y with respect to circle K. Find the length of segment RQ. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center Z and circumradius 5. Let N be the line through points E and F. Point V is the reflection of point A across line N. Point T is the reflection of point V across segment FD. What is the distance between points V and T? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJKL is a regular dodecagon with center T and circumradius 3. Let X be the circumcircle of polygon ABCDEFGHIJKL. Let Z be the reflection of point D across point J. Point W is the reflection of point J across point E. Line N passes through point L parallel to segment DZ. Circle V is the reflection of circle X across line N. Point Y is the inversion of point W with respect to circle V. What is the length of segment GY? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center T and circumradius 12. Let U be the midpoint of segment AJ. Let X be the reflection of point U across point L. Let O be the midpoint of segment JX. Point S is the midpoint of segment OF. Let Z be the reflection of point K across point E. Find the distance between points S and Z. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 10. Let X be the line through points A and B. Point Q is the reflection of point A across point B. Line U is the reflection of line X across point C. Line W passes through point H parallel to line U. Point P is the reflection of point J across line W. What is the length of segment PQ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center S and circumradius 3. Line J is the perpendicular bisector of segment FG. Line W passes through point A perpendicular to line J. Point Z is the reflection of point B across point E. Line M is the reflection of line W across point B. Point R is the reflection of point Z across line M. Point X is the midpoint of segment RD. What is the distance between points D and X? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJKL is a regular dodecagon with center P and circumradius 9. Let Z be the reflection of point B across point J. Let Q be the midpoint of segment DA. Point V is the midpoint of segment EQ. What is the length of segment ZV? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 3. Let Y be the perpendicular bisector of segment HE. Line S passes through point J parallel to line Y. Point R is the reflection of point M across line S. Let W be the reflection of point R across point D. What is the distance between points W and F? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJ be a regular 10-gon with center T and circumradius 10. Line Y passes through points D and A. Let V be the reflection of line Y through point B. Point X is the reflection of point T across line V. Line R passes through points I and X. Point U is the reflection of point J across line R. Find the distance between points U and T. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFHIJKLM is a regular dodecagon with center G and circumradius 4. Point V is the midpoint of segment KE. Line U is the angle bisector of angle IHV. Point W is the reflection of point E across line U. Point O is the reflection of point B across point W. What is the distance between points V and O? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABDEFGH is a regular 7-gon with center C and circumradius 8. Let Q be the circumcircle of polygon ABDEFGH. Point R is the midpoint of segment BH. Let I be the reflection of circle Q through point F. Point W is the inversion of point R with respect to circle I. What is the length of segment EW? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJL is a regular 11-gon with center K and circumradius 12. Let Q be the circumcircle of polygon ABCDEFGHIJL. Line S passes through points E and H. Let M be the line through point G perpendicular to line S. Let T be the reflection of circle Q across line M. Point X is the reflection of point H across point F. Let W be the inversion of point X in circle T. What is the length of segment JW? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABDEFGHI is a regular octagon with center C and circumradius 8. Let T be the perpendicular bisector of segment AC. Line R is the reflection of line T across point C. Line L is the reflection of line R across line T. Point Y is the reflection of point I across line L. Find the length of segment CY. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFHI be a regular octagon with center G and circumradius 3. Point R is the midpoint of segment IH. Point K is the midpoint of segment DR. Point O is the midpoint of segment CA. What is the distance between points O and K? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDFGHIJKLM is a regular dodecagon with center E and circumradius 5. Line O passes through point K parallel to segment HG. Line P passes through point C perpendicular to line O. Point Y is the reflection of point J across line P. Point X is the reflection of point Y across segment HG. Find the length of segment XK. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ACDEFG is a regular hexagon with center B and circumradius 3. Line I is the angle bisector of angle DBE. Point N is obtained by rotating point G by angle 5pi/12 about point B. Let M be the perpendicular bisector of segment AB. Point H lies on line I at distance 3 from point A. Let V be the line through point N perpendicular to line M. Line S passes through point H perpendicular to line V. Let P be the line through point D parallel to line S. Point Y is the reflection of point A across line P. Point Z is the reflection of point N across point G. Find the length of segment ZY. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let BCDEFGHI be a regular octagon with center A and circumradius 5. Let O be the line through points F and I. Let Q be the line through point H parallel to line O. Let R be the line through point B parallel to line O. Point S is the reflection of point C across line R. Point X is the reflection of point D across line Q. Find the distance between points S and X. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDE is a regular 5-gon with center Z and circumradius 9. Line I passes through points B and Z. Let H be the reflection of point D across point B. Let Y be the line through point B perpendicular to line I. Line P passes through points Z and C. Let T be the reflection of line Y through point H. Lines T and P intersect at point V. What is the length of segment VD? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center Y and circumradius 4. Line N is the perpendicular bisector of segment YB. Point Z is the reflection of point A across line N. Point T is the reflection of point Z across segment DY. Let H be the midpoint of segment ET. Point F is the reflection of point D across point H. Point G is the reflection of point D across segment EA. Find the length of segment GF. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center T and circumradius 11. Point X is the midpoint of segment BG. Line Q passes through points X and A. Point M is the intersection of line Q and segment FD. Let J be the reflection of point E across point H. Find the length of segment MJ. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center W and circumradius 12. Let R be the reflection of point G across point L. Let Y be the line through points D and G. Segment ER and line Y intersect at point Q. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJK is a regular 11-gon with center W and circumradius 10. Let V be the line through points E and F. Let L be the line through point C perpendicular to line V. Let P be the line through points E and A. Let S be the reflection of line P across line L. Point M is the reflection of point A across line S. Find the length of segment ME. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCD be a square with center M and circumradius 1. Line Z is the perpendicular bisector of segment DM. Let I be the midpoint of segment BA. Let H be the circumcircle of polygon ABCD. Let K be the reflection of circle H across line Z. Point X is the inversion of point I with respect to circle K. Find the length of segment XB. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ACDEF be a regular 5-gon with center B and circumradius 10. Point M is the midpoint of segment FB. Line W passes through point A perpendicular to segment CB. Point L is the reflection of point M across line W. Find the distance between points C and L. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFH be a regular 7-gon with center G and circumradius 4. Point W is obtained by rotating point D by angle 3pi/4 about point B. Let N be the reflection of point G across point D. Line Y is the perpendicular bisector of segment BC. Line K is the reflection of line Y across point N. Point O is the reflection of point W across line K. What is the length of segment NO? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center M and circumradius 12. Let X be the line through point G parallel to segment DH. Line L is the reflection of line X across point F. Point R is the reflection of point F across line L. Find the distance between points G and R. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJKM is a regular dodecagon with center L and circumradius 11. Let V be the line through points D and A. Point Q is obtained by rotating point M by angle pi/2 about point F. Let W be the circumcircle of polygon ABCDEFGHIJKM. Point X is the midpoint of segment JL. Circle U is the reflection of circle W across line V. Let P be the inversion of point H in circle U. Let R be the line through points P and X. Point T is the reflection of point C across line R. Find the distance between points T and Q. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJK is a regular 11-gon with center Z and circumradius 6. Line V passes through point A perpendicular to segment JD. Let W be the line through point H perpendicular to line V. Point S is the reflection of point D across line W. Find the distance between points S and Z. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
BCDEFGHIJK is a regular 10-gon with center A and circumradius 1. Let U be the line through points H and C. Let O be the midpoint of segment EA. Line V passes through point G parallel to line U. Point Z is the reflection of point O across line V. Point Y is the reflection of point A across segment DH. What is the length of segment YZ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
EY = 2. Let ABCDFGHIJ be a regular 9-gon with center Y and circumradius 9. Point N is on segment YJ. Let S be the circumcircle of polygon ABCDFGHIJ. Point M is the inversion of point N with respect to circle S. What is the measure of angle HYM, in radians? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDFGHI be a regular octagon with center E and circumradius 9. Let V be the angle bisector of angle ECG. Point O is the midpoint of segment IA. Point X is the reflection of point I across line V. Point N is the reflection of point H across point X. What is the length of segment NO? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 9. Point S is obtained by rotating point F by angle 7pi/12 about point A. Let Z be the rotation of point L by angle 5pi/6 around point G. Let R be the line through points M and D. Point X is the reflection of point Z across point D. Point N is the midpoint of segment SZ. Point Q is the reflection of point X across line R. What is the distance between points N and Q? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center L and circumradius 2. Let P be the circumcircle of polygon ABCDE. Line O is the angle bisector of angle LCA. Circle M is the reflection of circle P across line O. Let W be the perpendicular bisector of segment LC. Circle R is the reflection of circle M across point E. Point U is the inversion of point L with respect to circle R. Point G is the reflection of point U across line W. What is the distance between points G and B? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center H and circumradius 5. Let L be the vector from point C to point A. Point Q is the translation of point F by vector L. Let Z be the reflection of point Q across point D. Let K be the circumcircle of polygon ABCDEF. Let I be the midpoint of segment QA. Let W be the reflection of point D across point I. Let T be the reflection of circle K through point I. Let O be the midpoint of segment WE. Point R is the inversion of point Z with respect to circle T. Point J is the reflection of point I across segment OZ. What is the distance between points R and J? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center M and circumradius 12. Let L be the reflection of point F across point C. Let O be the reflection of point D across point B. Point Y is the midpoint of segment LM. What is the length of segment YO? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center V and circumradius 3. Let O be the angle bisector of angle LAI. Point N is the midpoint of segment HG. Let W be the perpendicular bisector of segment NV. Point M is the reflection of point J across line O. Let Y be the reflection of line W through point M. Point S is the reflection of point M across line Y. Find the distance between points S and C. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJK is a regular 11-gon with center X and circumradius 6. Let U be the line through points I and C. Point N is the reflection of point A across point H. Segment JD and line U intersect at point R. Find the length of segment RN. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDE is a regular 5-gon with center J and circumradius 8. Point X is the reflection of point A across point D. Let S be the circumcircle of polygon ABCDE. Let Z be the inversion of point X in circle S. Point T is the midpoint of segment AD. What is the length of segment ZT? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFG is a regular 7-gon with center I and circumradius 10. Line Q is the perpendicular bisector of segment IC. Line T is the reflection of line Q across point G. Line P passes through points D and A. Lines T and P intersect at point M. Let L be the perpendicular bisector of segment MI. Point K is the reflection of point A across line L. What is the distance between points K and A? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center W and circumradius 11. Let O be the line through points D and B. Point X is the reflection of point H across line O. Point S is the reflection of point F across line O. Point K is the reflection of point S across segment WX. What is the length of segment AK? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
FU = 3. Let ABCDE be a regular 5-gon with center U and circumradius 2. Point I is the midpoint of segment CD. Point R is the midpoint of segment IC. Point O is the midpoint of segment IR. Point G is the reflection of point D across point E. Find the length of segment GO. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJ is a regular 10-gon with center P and circumradius 3. Let V be the reflection of point D across point F. Line X passes through point G perpendicular to segment BV. Point S is the reflection of point J across line X. What is the distance between points E and S? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center L and circumradius 8. Let H be the midpoint of segment FD. Let K be the midpoint of segment DC. Point Z is the reflection of point K across segment BH. What is the measure of angle BAZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center P and circumradius 1. Let K be the midpoint of segment GP. Line X is the perpendicular bisector of segment AP. Point N is obtained by rotating point A by angle 2pi/3 about point P. Point T is the reflection of point D across point H. Line J passes through points K and E. Line Z passes through point T perpendicular to line J. Let W be the reflection of line Z across line X. Let V be the reflection of line W through point F. Line V and segment BN intersect at point O. Point L is the reflection of point O across line X. What is the length of segment LP? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCD is a square with center N and circumradius 9. Line Q is the perpendicular bisector of segment DC. Line J is the reflection of line Q across point D. Point T is obtained by rotating point N by angle 5pi/6 about point A. Point H is the reflection of point T across line J. Let P be the line through points N and H. Let U be the midpoint of segment DH. Point E is the reflection of point D across line P. Find the length of segment EU. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center P and circumradius 1. Let W be the circumcircle of polygon ABCDEF. Point Q is the midpoint of segment DF. Point V is the inversion of point Q with respect to circle W. Point O is the reflection of point B across point F. What is the length of segment VO? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center R and circumradius 7. Line Y is the perpendicular bisector of segment ER. Let O be the midpoint of segment AR. Let K be the line through point D parallel to line Y. Point J is the reflection of point O across line K. Let T be the midpoint of segment OD. Let G be the reflection of point R across point T. Find the distance between points J and G. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCD be a square with center Z and circumradius 12. Let X be the angle bisector of angle DBC. Point Q is the midpoint of segment DC. Line O is the perpendicular bisector of segment CQ. Point J is the reflection of point Z across line O. Point W is the reflection of point A across line X. What is the distance between points J and W? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ACDE be a square with center B and circumradius 10. Let W be the rotation of point E by angle 2pi/3 around point A. Let X be the circumcircle of polygon ACDE. Let K be the line through points D and B. Let Y be the line through points C and W. Let J be the midpoint of segment DE. Let Q be the reflection of line Y across line K. Lines K and Q intersect at point R. Let T be the inversion of point J in circle X. What is the length of segment TR? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDE is a regular 5-gon with center N and circumradius 6. Let W be the rotation of point E by angle pi/2 around point B. Let Z be the perpendicular bisector of segment CW. Let S be the line through point N parallel to line Z. Line M passes through points C and E. Let H be the reflection of line S across line M. Point X is the reflection of point N across line H. Find the distance between points X and D. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJ is a regular 10-gon with center O and circumradius 4. Line Z passes through point D perpendicular to segment OI. Let N be the line through point H parallel to line Z. Point Q is the reflection of point G across line N. What is the measure of angle ACQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center J and circumradius 7. Let Q be the circumcircle of polygon ABCDEF. Point X is obtained by rotating point J by angle pi/12 about point F. Point M is the reflection of point E across segment CB. Point H is the inversion of point X with respect to circle Q. Point T is the reflection of point E across segment HB. What is the length of segment TM? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center V and circumradius 6. Let Y be the circumcircle of polygon ABCDEF. Let K be the reflection of circle Y through point B. Point H is the reflection of point E across point A. Let M be the reflection of circle K through point H. Point Q is the inversion of point E with respect to circle M. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ACDEFGH is a regular 7-gon with center B and circumradius 10. Line Y is the perpendicular bisector of segment GB. Let Q be the circumcircle of polygon ACDEFGH. Let V be the reflection of circle Q through point E. Point R is the reflection of point A across line Y. Point W is the inversion of point B with respect to circle V. What is the distance between points W and R? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center N and circumradius 4. Point V is the midpoint of segment CE. Let Y be the reflection of point F across point E. Point K is the reflection of point N across point D. Point R is the reflection of point K across point E. Line M passes through points V and F. Let O be the line through point R parallel to line M. Point S is the reflection of point Y across line O. What is the length of segment SN? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center P and circumradius 3. Let G be the reflection of point E across point C. Point S is the reflection of point G across segment DE. Point Z is the reflection of point S across point D. Point W is the reflection of point Z across segment DE. What is the length of segment WE? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCD be a square with center H and circumradius 3. Line Y is the perpendicular bisector of segment AB. Let I be the line through point D perpendicular to line Y. Point V is the reflection of point H across line I. Point O is the reflection of point H across segment BV. Point J is the midpoint of segment BV. Let N be the midpoint of segment DJ. Find the length of segment ON. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 7. Let P be the line through points E and G. Let N be the line through point A parallel to line P. Let Z be the midpoint of segment LK. Point Q is the reflection of point Z across line N. What is the length of segment KQ? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCD be a square with center T and circumradius 11. Line O passes through points T and A. Point F is the midpoint of segment AD. Point Y is the reflection of point B across point D. Point S is the reflection of point Y across line O. What is the length of segment SF? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIJ be a regular 10-gon with center P and circumradius 3. Let S be the midpoint of segment FB. Let M be the angle bisector of angle HAI. Let Q be the line through point C parallel to line M. Line R passes through point B perpendicular to line Q. Let Z be the line through point S perpendicular to line R. Point W is the reflection of point A across line Z. What is the distance between points E and W? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center U and circumradius 4. Let N be the circumcircle of polygon ABCDEF. Line L passes through points E and D. Let O be the line through point F perpendicular to line L. Circle W is the reflection of circle N across line O. Let I be the inversion of point D in circle W. What is the measure of angle AID, in radians? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center H and circumradius 10. Let X be the perpendicular bisector of segment HD. Let U be the reflection of line X through point E. Point W is the reflection of point H across line U. Point T is the midpoint of segment EC. Let N be the midpoint of segment WD. What is the distance between points N and T? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center S and circumradius 7. Line I is the angle bisector of angle ACB. Line K is the reflection of line I across point S. Let V be the line through points B and C. Line G is the reflection of line K across line V. Point U is the reflection of point C across line G. Find the distance between points U and D. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCD be a square with center I and circumradius 2. Line O is the angle bisector of angle DCI. Let M be the reflection of line O through point A. Point G is the reflection of point D across line M. Let V be the reflection of point B across point D. Let J be the midpoint of segment CG. Point K is the midpoint of segment DJ. Find the distance between points V and K. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCD is a square with center E and circumradius 11. Point F is the reflection of point A across point D. Point M is the reflection of point C across point B. Let W be the perpendicular bisector of segment BD. Let X be the line through point M parallel to line W. Point N is the reflection of point F across line X. What is the distance between points N and C? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDF is a regular 5-gon with center E and circumradius 10. Let X be the midpoint of segment EA. Line J passes through point C parallel to segment FB. Point T is obtained by rotating point X by angle pi/2 about point D. Point M is the reflection of point A across point C. Point P is the reflection of point T across line J. Point Z is the reflection of point M across segment FB. Let I be the line through points Z and P. Point U is the reflection of point X across line I. What is the length of segment UD? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center M and circumradius 1. Point Q is the reflection of point B across segment DF. Line Z is the perpendicular bisector of segment DB. Let T be the perpendicular bisector of segment FQ. Let R be the reflection of line Z across line T. Point H is the reflection of point M across line R. Find the distance between points F and H. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGH is a regular octagon with center M and circumradius 5. Let T be the angle bisector of angle AGB. Point X is the reflection of point C across line T. Point N is the reflection of point X across segment AF. Find the distance between points N and H. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center M and circumradius 4. Let U be the circumcircle of polygon ABCDEF. Line I is the perpendicular bisector of segment MF. Let W be the reflection of circle U across line I. Let O be the inversion of point C in circle W. Point R is the reflection of point D across line I. Find the length of segment RO. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFH is a regular 7-gon with center G and circumradius 5. Let X be the rotation of point D by angle 3pi/4 around point B. Let O be the midpoint of segment XB. Let J be the angle bisector of angle FOA. Line M is the reflection of line J across point F. Point I is the reflection of point H across line M. What is the length of segment GI? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ACDEF be a regular 5-gon with center B and circumradius 10. Point P is obtained by rotating point C by angle pi/3 about point A. Line W is the perpendicular bisector of segment FA. Point V is the reflection of point P across segment FA. Let Z be the line through point V parallel to line W. Point H is the intersection of line Z and segment FA. What is the distance between points V and H? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHJKLM be a regular dodecagon with center I and circumradius 10. Line V passes through points E and C. Let U be the angle bisector of angle JDB. Line S is the reflection of line U across line V. Lines S and V intersect at point Z. Find the length of segment ZI. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center X and circumradius 8. Let U be the angle bisector of angle HAD. Point T is the reflection of point E across point D. Point R is the reflection of point T across line U. Line Y passes through point R parallel to segment GF. Line K passes through point A parallel to line Y. Point N is the reflection of point R across line K. What is the length of segment NT? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 12. Let R be the circumcircle of polygon ABCDEFGHIJKL. Let N be the midpoint of segment EJ. Let M be the reflection of circle R through point K. Point O is the reflection of point N across point L. Point Q is the inversion of point O with respect to circle M. What is the distance between points L and Q? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABDEFG be a regular hexagon with center C and circumradius 6. Point Q is obtained by rotating point F by angle pi/12 about point E. Let Z be the angle bisector of angle BAE. Let T be the line through point Q perpendicular to line Z. Point H is the reflection of point C across line T. Line P is the angle bisector of angle HCQ. Point N is the reflection of point G across line P. What is the length of segment NH? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center I and circumradius 7. Point Q is the midpoint of segment AC. Point R is the midpoint of segment QD. Point K is the midpoint of segment HF. What is the distance between points K and R? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFG is a regular 7-gon with center H and circumradius 6. Let J be the line through points E and C. Point R is the reflection of point F across line J. Point Y is the reflection of point G across segment CB. What is the measure of angle RYB, in radians? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFG is a regular 7-gon with center R and circumradius 12. Line J passes through points B and D. Line P passes through point R perpendicular to segment FD. Let T be the reflection of line P across line J. Point Z is the intersection of lines J and T. Point L is the midpoint of segment FD. What is the distance between points L and Z? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center U and circumradius 2. Let I be the reflection of point A across point C. Point Q is the reflection of point E across point I. Let J be the midpoint of segment ID. Line V passes through point Q perpendicular to segment CJ. Point Z is the reflection of point C across line V. What is the distance between points I and Z? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center P and circumradius 3. Let Q be the rotation of point B by angle 3pi/4 around point C. Point U is obtained by rotating point E by angle pi about point A. Let S be the rotation of point Q by angle 5pi/6 around point U. Let Z be the midpoint of segment AF. Let T be the line through point U perpendicular to segment DU. Point L is the reflection of point Z across line T. What is the distance between points S and L? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center L and circumradius 11. Point S is obtained by rotating point D by angle 5pi/6 about point C. Line I passes through point C perpendicular to segment BS. Line R passes through point A parallel to line I. Point N is the reflection of point C across line R. What is the length of segment NE? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJK is a regular 11-gon with center P and circumradius 4. Line N passes through points B and D. Point R is the intersection of segment FC and line N. Let M be the midpoint of segment PI. Find the length of segment MR. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center W and circumradius 1. Let M be the rotation of point C by angle 5pi/12 around point E. Let R be the perpendicular bisector of segment MF. Let J be the line through point W perpendicular to segment EB. Let S be the reflection of line J across line R. Line S and segment MB intersect at point Y. What is the length of segment YC? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center H and circumradius 6. Point Q is the reflection of point A across point F. Let U be the reflection of point B across point Q. Let L be the circumcircle of polygon ABCDEF. Point P is the inversion of point U with respect to circle L. Find the length of segment PD. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center W and circumradius 1. Point U is the reflection of point D across point W. Point O is the midpoint of segment CD. Let K be the line through points B and E. Point P is the reflection of point D across point U. Line R is the reflection of line K across point O. Let S be the reflection of point P across point O. Point V is the reflection of point P across line R. Point G is the reflection of point S across point V. Find the length of segment UG. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center P and circumradius 3. Point V is the reflection of point F across point B. Line Q passes through points V and P. Point M is the reflection of point B across line Q. Point Y is the midpoint of segment PA. Find the length of segment MY. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center M and circumradius 7. Let N be the reflection of point E across point A. Let K be the circumcircle of polygon ABCDEF. Line I is the angle bisector of angle MNC. Let X be the reflection of circle K across line I. Let G be the inversion of point F in circle X. What is the length of segment NG? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEF be a regular hexagon with center I and circumradius 11. Let T be the circumcircle of polygon ABCDEF. Let P be the reflection of circle T through point D. Point K is the inversion of point A with respect to circle P. Point J is the reflection of point B across point F. Find the distance between points K and J. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFHIJ be a regular 9-gon with center G and circumradius 10. Point Q is the midpoint of segment FG. Let L be the line through points I and E. Line X passes through point D perpendicular to line L. Point R is the reflection of point Q across line X. Point P is the reflection of point R across line L. What is the distance between points P and B? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHIJ is a regular 10-gon with center X and circumradius 7. Let V be the midpoint of segment XC. Point O is the reflection of point X across point F. Let Z be the circumcircle of polygon ABCDEFGHIJ. Let Q be the inversion of point V in circle Z. Let T be the reflection of point O across point Q. Find the length of segment BT. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 1. Let N be the angle bisector of angle GDJ. Point S is the reflection of point B across line N. Line O is the angle bisector of angle CSF. Point R is the reflection of point A across line O. Find the length of segment SR. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFGHI is a regular 9-gon with center O and circumradius 12. Let W be the circumcircle of polygon ABCDEFGHI. Line V passes through points C and D. Circle S is the reflection of circle W across line V. Point J is the inversion of point G with respect to circle S. Find the distance between points A and J. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDEFGH be a regular octagon with center S and circumradius 11. Let N be the midpoint of segment CA. Point M is the midpoint of segment CF. Line X passes through point E perpendicular to segment FN. Let P be the reflection of point H across point M. Point R is the reflection of point D across line X. Find the length of segment PR. Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
Let ABCDE be a regular 5-gon with center F and circumradius 2. Let L be the rotation of point B by angle pi/6 around point A. Let G be the reflection of point B across point F. Let N be the vector from point B to point G. Line W is the perpendicular bisector of segment EC. Point V is the translation of point F by vector N. Let R be the reflection of point V across point C. Let Z be the line through point D perpendicular to line W. Line I passes through point L parallel to line Z. Line I and segment CR intersect at point U. Let O be the reflection of point U across point D. What is the length of segment UO? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEFG is a regular 7-gon with center R and circumradius 12. Point Z is the reflection of point A across segment GB. Let K be the angle bisector of angle BAZ. Point W is the reflection of point D across line K. What is the distance between points W and G? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}


[User]
ABCDEF is a regular hexagon with center O and circumradius 3. Line T passes through points A and F. Let Q be the perpendicular bisector of segment EF. Let J be the midpoint of segment EF. Let Y be the circumcircle of polygon ABCDEF. Line R is the reflection of line T across line Q. Circle S is the reflection of circle Y across line R. Let H be the inversion of point J in circle S. What is the length of segment OH? Express your answer as a decimal number rounded to 2 decimal places.

Present the answer in LaTex format: \boxed{Your answer}