[User]
ABCDE is a regular 5-gon with center G and circumradius 6. Let P be the reflection of point B across point G. Let F be the perpendicular bisector of segment DA. Line Y passes through point B parallel to line F. Point Q is the reflection of point D across line Y. Line X is the reflection of line F across line Y. Let K be the reflection of line X across line F. Point O is the reflection of point P across line K. What is the length of segment QO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ACDEFGHIJK is a regular 10-gon with center B and circumradius 3. Line Y passes through points E and H. Let N be the midpoint of segment GD. Line R is the reflection of line Y across point N. Point L is the intersection of line R and segment IH. Find the distance between points B and L. 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 X and circumradius 12. Point K is obtained by rotating point X by angle pi/3 about point F. Let I be the midpoint of segment KG. Point J is the midpoint of segment EI. Let L be the line through points E and J. Let W be the midpoint of segment KX. Let U be the line through points G and W. Lines L and U intersect at point H. Point V is the midpoint of segment XD. Find the length of segment HV. 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 R and circumradius 9. Let W be the reflection of point B across point D. Point P is the midpoint of segment AR. Point V is the midpoint of segment IP. Point Y is the reflection of point V across point R. What is 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 ABCDEFGHIJKL be a regular dodecagon with center O and circumradius 6. Line Y is the angle bisector of angle EHG. Line M passes through point D parallel to line Y. Point Z is the intersection of segment CH and line M. Find the distance between points Z 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 ABCDE be a regular 5-gon with center Z and circumradius 10. Let M be the circumcircle of polygon ABCDE. Point Q is the reflection of point Z across point D. Let R be the midpoint of segment BQ. Let I be the inversion of point R in circle M. What is the distance between points R and I? 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 K and circumradius 10. Point Q is the reflection of point F across point B. Let L be the rotation of point I by angle 5pi/12 around point A. Let Y be the circumcircle of polygon ABCDEFGHI. Point V is the inversion of point Q with respect to circle Y. Let J be the perpendicular bisector of segment FL. Line Z passes through point F parallel to line J. Line S passes through point V parallel to line Z. Line S and segment FL intersect at point R. What is the distance between points R and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFGHI is a regular octagon with center A and circumradius 10. Point L is the midpoint of segment EG. Line M is the angle bisector of angle DGL. Point K is the reflection of point D across line M. Point J is the reflection of point K across point H. Point U is the midpoint of segment JL. What is the distance between points U and C? 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 1. Point K is the reflection of point D across segment BC. Point I is the midpoint of segment KG. Point X is the midpoint of segment KC. Find the length of segment IX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDFGHIJKL be a regular 11-gon with center E and circumradius 4. Let S be the circumcircle of polygon ABCDFGHIJKL. Line Z is the angle bisector of angle CKG. Line N is the perpendicular bisector of segment EJ. Let Y be the midpoint of segment AC. Let T be the reflection of line N across line Z. Circle X is the reflection of circle S across line T. Let M be the inversion of point Y in circle X. Find the length of segment MG. 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 T and circumradius 4. Point X is the midpoint of segment DE. Let Q be the rotation of point D by angle pi/12 around point A. Let M be the angle bisector of angle TXQ. Let O be the reflection of line M through point F. Point I is the reflection of point B across line O. Point V is the intersection of segment XI and line O. What is the length of segment VA? 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 R and circumradius 11. Let Z be the circumcircle of polygon ABCDEFGHIJKL. Let Q be the midpoint of segment IL. Let P be the inversion of point Q in circle Z. What is the distance between points Q and P? 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 2. Line J is the angle bisector of angle DBF. Let V be the reflection of line J through point A. Point T is the reflection of point C across point F. Line X passes through points T and C. Point M is the reflection of point C across line V. Point L is the reflection of point D across line X. Find the length of segment LM. 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 9. Point Y is obtained by rotating point A by angle pi/6 about point F. Line Q is the angle bisector of angle DLK. Let T be the reflection of line Q through point K. Point M is the reflection of point F across line T. Point O is the midpoint of segment MA. Find the length of segment OY. 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 5. Line Z is the angle bisector of angle BCF. Line S passes through point R perpendicular to line Z. Point V is the reflection of point F across line S. Point X is the midpoint of segment RD. Point I is the midpoint of segment VX. What is the distance between points I 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 ABCDEFGHIJKL be a regular dodecagon with center Q and circumradius 1. Let R be the perpendicular bisector of segment QE. Line T is the reflection of line R across point A. Line S is the reflection of line T across point J. Line S and segment DJ intersect at point X. What is the length of segment XC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFG is a regular hexagon with center A and circumradius 2. Point L is the midpoint of segment BF. Line N is the perpendicular bisector of segment LE. Point I is the reflection of point B across line N. Find the length of segment AI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ACDEFGHI is a regular octagon with center B and circumradius 9. Let K be the midpoint of segment HI. Point O is the reflection of point K across point G. Point L is the midpoint of segment BI. Point T is the midpoint of segment LD. Find 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]
Let ABCDEFGHIJKL be a regular dodecagon with center W and circumradius 4. Let X be the rotation of point G by angle 7pi/12 around point B. Let U be the line through point I parallel to segment FL. Let T be the circumcircle of polygon ABCDEFGHIJKL. Line Z passes through point A parallel to segment FL. Let V be the reflection of line U through point W. Let M be the reflection of circle T across line V. Circle R is the reflection of circle M across line Z. Point Y is the inversion of point X with respect to circle R. What is the length of segment YA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJLM is a regular dodecagon with center K and circumradius 3. Point Z is the reflection of point M across point H. Let R be the line through points L and H. Let Q be the midpoint of segment ZA. Point N is the reflection of point G across line R. Find the length of segment NQ. 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 M and circumradius 9. Line Y is the angle bisector of angle LAD. Let W be the perpendicular bisector of segment CE. Let X be the reflection of line Y across line W. Line V passes through point M parallel to line Y. Point O is the intersection of lines V and X. Find the length of segment DO. 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 5. Let M be the perpendicular bisector of segment AF. Point N is the reflection of point F across point E. Point V is the reflection of point N across line M. Find the length of segment CV. 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 8. Point U is the reflection of point C across segment MG. Segments GJ and IL intersect at point Q. Let X be the perpendicular bisector of segment AL. Point O is the reflection of point Q across line X. What is the distance between points O and U? 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 8. Line G passes through points A and C. Let V be the line through point D parallel to line G. Line Y is the reflection of line G across line V. Let U be the line through points B and N. Let S be the reflection of line Y through point N. Point R is the intersection of lines S and U. Find the measure of angle DNR, 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 T and circumradius 9. Point M is the midpoint of segment HB. Point X is the midpoint of segment GE. Line P passes through point X perpendicular to segment DH. Line L is the reflection of line P across point E. Point Q is the reflection of point M across line L. Find the length of segment QF. 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 4. Let Y be the line through points B and K. Line Z is the reflection of line Y across point C. Point U is the intersection of line Z and segment HB. What is the distance between points U 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 6. Point H is the midpoint of segment YE. Point N is obtained by rotating point H by angle pi/2 about point A. Point V is the midpoint of segment NB. Let P be the midpoint of segment DN. Point R is the reflection of point V across point C. Find the distance between points R 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 ABCDEF be a regular hexagon with center I and circumradius 4. Point J is the reflection of point D across point C. Let R be the circumcircle of polygon ABCDEF. Point V is the inversion of point J with respect to circle R. Let H be the reflection of point I across point C. Find the measure of angle HBV, 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 U and circumradius 11. Line W passes through points C and B. Point N is the reflection of point G across segment BF. Point L is the reflection of point A across line W. Find the length of segment LN. 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 J and circumradius 10. Let U be the rotation of point C by angle 11pi/12 around point F. Point X is the midpoint of segment DB. Point W is the midpoint of segment FU. 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]
Let ABCDEFGHIJKL be a regular dodecagon with center R and circumradius 12. Let N be the rotation of point D by angle 2pi/3 around point F. Point V is the reflection of point K across segment ND. Let S be the angle bisector of angle LVK. Point O is the reflection of point L across line S. What is 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]
ABCDEGHIJ is a regular 9-gon with center F and circumradius 12. Line V passes through points H and C. Line S is the angle bisector of angle FJB. Line M is the reflection of line S across line V. Point P is the reflection of point F across line M. Let Q be the reflection of point P across point G. Find the length of segment AQ. 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 W and circumradius 7. Point O is the midpoint of segment WD. Point K is the reflection of point O across segment WA. Let Z be the perpendicular bisector of segment WA. Point V is the reflection of point C across line Z. Find 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]
ABCDEFGHIJ is a regular 10-gon with center Z and circumradius 11. Let L be the circumcircle of polygon ABCDEFGHIJ. Circle Q is the reflection of circle L across point I. Point X is the inversion of point G with respect to circle Q. Point M is the reflection of point B across point H. What is the length of segment XM? 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 P and circumradius 12. Let R be the circumcircle of polygon ABCD. Line Z is the angle bisector of angle DAP. Line N is the perpendicular bisector of segment BP. Let I be the reflection of circle R across line Z. Let H be the reflection of circle I across line N. Let S be the perpendicular bisector of segment AP. Point T is the inversion of point C with respect to circle H. Point L is the reflection of point T across line S. What is the distance between points T and L? 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 6. Point U is obtained by rotating point H by angle pi/6 about point B. Let W be the angle bisector of angle UDJ. Let Z be the perpendicular bisector of segment DC. Let N be the reflection of line W across line Z. Let P be the line through point U parallel to line N. Point X is the reflection of point K across line P. 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]
BCDEFG is a regular hexagon with center A and circumradius 10. Line H passes through points D and A. Let L be the reflection of line H through point C. Let W be the reflection of line L across line H. Let O be the reflection of point E across point B. Point Y is the reflection of point O across line W. Find the distance between points E and Y. 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. Point Y is the reflection of point H across point D. Let K be the midpoint of segment FD. Point Z is the midpoint of segment GY. Let Q be the reflection of point A across point Z. 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 ABCDEFGH be a regular octagon with center P and circumradius 7. Line I is the angle bisector of angle PDH. Line M passes through points E and C. Point V is the reflection of point F across line M. Point N is the reflection of point V across line I. What is the distance between points N 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 ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 8. Line Y is the perpendicular bisector of segment LK. Line X is the reflection of line Y across point A. Let R be the reflection of line X across line Y. Point Z is the reflection of point E across line X. Point S is the reflection of point F across line X. Line Q is the angle bisector of angle ZMS. Line P is the reflection of line R across line Q. Let V be the line through points I and C. Lines P and V intersect at point T. Find the length of segment ST. 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 M and circumradius 12. Let R be the reflection of point C across point F. Let Q be the midpoint of segment GF. Let P be the line through points Q and J. Segment DI and line P intersect at point Y. What is the length of segment YR? 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 O and circumradius 12. Point N is the reflection of point D across point B. Let Y be the perpendicular bisector of segment CA. Point M is the reflection of point D across segment AN. Point K is the reflection of point M across line Y. Let J be the midpoint of segment MK. Find the distance between points J 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 ABCDEFGHIJKL be a regular dodecagon with center T and circumradius 10. Let Q be the midpoint of segment LA. Let O be the line through point F parallel to segment GK. Point V is the reflection of point Q across line O. Find the length of segment VL. 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 R and circumradius 5. Let O be the perpendicular bisector of segment LR. Line U is the reflection of line O across point C. Let M be the circumcircle of polygon ABCDEFGHIJKL. Let N be the reflection of circle M across line U. Point X is the inversion of point J with respect to circle N. Find the length of segment XG. 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 S and circumradius 3. Let W be the circumcircle of polygon ABCDEFGHIJKL. Let Z be the angle bisector of angle HCD. Circle Y is the reflection of circle W across line Z. Line T is the perpendicular bisector of segment SK. Let P be the rotation of point D by angle 2pi/3 around point F. Circle N is the reflection of circle Y across line T. Point X is the midpoint of segment PJ. Let U be the inversion of point X in circle N. Point M is the reflection of point U across line Z. Find the distance between points M 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 ABCDFGHIJ be a regular 9-gon with center E and circumradius 10. Let N be the rotation of point A by angle 11pi/12 around point J. Point L is the reflection of point B across point N. Let S be the angle bisector of angle GCJ. Point Y is the reflection of point L across line S. Point Z is the reflection of point H across point Y. 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]
EM = 8. ABCDFGHIJKL is a regular 11-gon with center M and circumradius 7. Line O is the perpendicular bisector of segment JI. Point Z is the midpoint of segment GC. Line T passes through point D perpendicular to line O. Line V passes through point K perpendicular to line T. Point Y is the reflection of point Z across line V. Find the length of segment LY. 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 8. Let L be the line through points G and I. Let T be the midpoint of segment BG. Point S is the reflection of point D across point F. Let Z be the perpendicular bisector of segment BT. Let V be the line through points S and F. Let Q be the reflection of line Z across line V. Lines L and Q intersect at point N. Point M is the midpoint of segment FE. What is the length of segment MN? 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 Q and circumradius 8. Let X be the reflection of point Q across point A. Let H be the angle bisector of angle QXB. Point O is the reflection of point C across line H. Let M be the midpoint of segment EO. What is the distance between points M 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 X and circumradius 2. Let Y be the line through points C and G. Line K passes through points H and X. Let S be the reflection of line Y through point E. Lines K and S intersect at point Q. What is the length of segment QA? 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 R and circumradius 1. Let J be the line through points B and D. Segment CF and line J intersect at point W. Let N be the midpoint of segment AW. Find the length of segment FN. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 5. Let Q be the circumcircle of polygon ABCDEFGIJKLM. Let X be the midpoint of segment MJ. Let S be the reflection of circle Q through point D. Point R is the reflection of point K across point X. Let Z be the inversion of point E in circle S. Point N is the reflection of point F across segment RK. What is the length of segment NZ? 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 Q and circumradius 9. Let Z be the circumcircle of polygon ABCD. Let O be the midpoint of segment BA. Let V be the reflection of circle Z through point B. Point W is the inversion of point O with respect to circle V. Point M is the reflection of point W across point A. What is the distance between points M 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 BCDEFGHI be a regular octagon with center A and circumradius 3. Point U is the reflection of point C across point I. Let K be the midpoint of segment FA. Let X be the line through point K parallel to segment BE. Point P is the reflection of point H across line X. Find the length of segment UP. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEF is a regular 5-gon with center D and circumradius 2. Let K be the rotation of point B by angle 11pi/12 around point C. Point M is the reflection of point C across point K. Point T is the reflection of point F across point E. Point O is the reflection of point M across segment DK. What is the length of segment TO? 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 U and circumradius 10. Line Z passes through points D and C. Let X be the line through point E parallel to line Z. Line Q is the reflection of line X across line Z. Point G is the midpoint of segment CD. Point V is the reflection of point G across line Q. Find the length of segment VC. 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 11. Let X be the reflection of point K across point J. Let Z be the perpendicular bisector of segment IK. Line R passes through points M and X. Let Y be the reflection of line R across line Z. Point S is the reflection of point X across line Y. Find the distance between points A 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 ABCDEFGHIJK be a regular 11-gon with center Q and circumradius 8. Line N is the perpendicular bisector of segment GF. Line O is the angle bisector of angle JQA. Let P be the line through point I parallel to line O. Line Z passes through point C parallel to line P. Let L be the reflection of line Z across line N. Lines Z and L intersect at point U. What is the measure of angle UQG, 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 Q and circumradius 8. Line Z is the angle bisector of angle DFE. Let L be the perpendicular bisector of segment DE. Let Y be the rotation of point F by angle pi/12 around point A. Let I be the reflection of line Z across line L. Let R be the reflection of line L across line I. Point J is the reflection of point E across point D. Line V passes through point D parallel to line R. Let G be the angle bisector of angle DJF. Let O be the line through point Y parallel to line G. Point K is the intersection of lines V and O. What is the distance between points E and K? 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 S and circumradius 9. Line T is the angle bisector of angle CAF. Point U is the reflection of point H across line T. Let Z be the circumcircle of polygon ABCDEFGH. Circle L is the reflection of circle Z across point U. Point X is the inversion of point E with respect to circle L. Find the distance between points X 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 ABCDEFG be a regular 7-gon with center Q and circumradius 9. Let M be the circumcircle of polygon ABCDEFG. Let P be the reflection of circle M through point F. Point H is the inversion of point A with respect to circle P. Point J is the inversion of point G with respect to circle P. Point T is the reflection of point J across point G. Find the length of segment TH. 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 P and circumradius 10. Point O is obtained by rotating point C by angle pi about point B. Line R passes through point C perpendicular to segment CG. Line Y passes through point E parallel to line R. Let Q be the circumcircle of polygon ABCDEFG. Let W be the inversion of point O in circle Q. Let H be the reflection of circle Q through point W. Line L passes through points D and P. Point U is the reflection of point A across point G. Let I be the reflection of line Y across line L. Circle Z is the reflection of circle H across line I. Let S be the inversion of point U in circle Z. Find the distance between points S and U. 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 Y and circumradius 4. Let M be the circumcircle of polygon ABCDEFGHIJ. Circle V is the reflection of circle M across point G. Let N be the reflection of point C across point B. Point L is the inversion of point N with respect to circle V. What is the length of segment IL? 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 4. Let U be the perpendicular bisector of segment CD. Line T is the reflection of line U across point D. Let N be the line through point I perpendicular to line T. Point Q is the reflection of point A across line N. Find the measure of angle BQH, 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 ABCD be a square with center W and circumradius 10. Line H is the angle bisector of angle DBW. Line Y passes through points W and A. Line N passes through point A parallel to line H. Point U is the midpoint of segment WC. Point F is the reflection of point B across line N. Point M is the reflection of point F across line Y. What is the distance between points U and M? 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 I and circumradius 8. Line P is the perpendicular bisector of segment CI. Point Q is the reflection of point D across line P. Let N be the midpoint of segment DQ. Point R is the midpoint of segment QC. 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]
BCDEFGHIJKLM is a regular dodecagon with center A and circumradius 12. Let S be the line through points M and J. Line Z is the perpendicular bisector of segment GF. Let T be the reflection of line Z through point E. Lines S and T intersect at point Y. What is the distance between points F 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 W and circumradius 12. Point N is on segment EG. Let Y be the perpendicular bisector of segment GB. Point L is the reflection of point N across line Y. What is the measure of angle ENL, 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 X and circumradius 10. Let S be the angle bisector of angle XHE. Let J be the midpoint of segment FA. Point N is the reflection of point X across line S. Point T is the midpoint of segment NJ. Find the distance between points T and C. 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 G and circumradius 11. Let F be the rotation of point A by angle pi/3 around point G. Point K is the midpoint of segment DF. Point X is the reflection of point B across point K. Point O is the midpoint of segment FX. Point H is the reflection of point O across point X. What is the length of segment KH? 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 W and circumradius 9. Let I be the circumcircle of polygon ABCD. Let T be the reflection of circle I through point D. Let X be the midpoint of segment WB. Point G is the inversion of point X with respect to circle T. Find the length of segment GW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHJK be a regular 10-gon with center I and circumradius 4. Line O passes through points H and B. Let P be the line through point K parallel to segment JE. Lines P and O intersect at point S. What is the length of segment SG? 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 S and circumradius 11. Let N be the perpendicular bisector of segment ID. Point M is the midpoint of segment CD. Let X be the line through point M parallel to line N. Segment ID and line X intersect at point Y. Find the distance between points Y 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 ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 7. Let Z be the midpoint of segment DC. Point V is the reflection of point Z across segment MF. Find 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]
ABCDEFG is a regular 7-gon with center R and circumradius 11. Let O be the line through points A and C. Point Y is the reflection of point F across line O. Point V is the reflection of point Y across segment BC. What is the distance between points V and B? 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 X and circumradius 10. Point Y is the reflection of point G across point E. Let P be the midpoint of segment XA. Point S is obtained by rotating point P by angle 3pi/4 about point F. Point M is the reflection of point E across point P. Line R passes through points M and J. Let Z be the line through point Y parallel to line R. Let V be the reflection of line Z through point D. Let N be the midpoint of segment SL. Point T is the reflection of point N across line V. What is the distance between points T and I? 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 2. Let X be the line through points I and G. Let Q be the angle bisector of angle KFM. Let Y be the reflection of line X across line Q. Point R is the reflection of point G across line Y. What is the distance between points M and R? 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 11. Let K be the circumcircle of polygon ABCDEFGH. Let Z be the reflection of circle K through point B. Point J is the midpoint of segment FA. Point Q is the inversion of point I with respect to circle Z. Point O is the midpoint of segment CJ. What is the distance between points O and Q? 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 3. Line R passes through points C and D. Line Y passes through point G parallel to segment HF. Point P is the intersection of lines R and Y. What is the length of segment HP? 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 Y and circumradius 4. Line K is the perpendicular bisector of segment YF. Point I is the reflection of point C across line K. Let U be the line through points G and A. Let W be the line through point I perpendicular to line U. Point O is the reflection of point Y across line W. Find the distance between points O and Y. 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 Z and circumradius 5. Line P passes through point B perpendicular to segment GE. Point M lies on line P. Let W be the reflection of point F across point D. Find the measure of angle WBM, 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 ABCDEFGHIJK be a regular 11-gon with center T and circumradius 7. Let N be the perpendicular bisector of segment FI. Point Z is the reflection of point A across point C. Line R is the reflection of line N across point I. Line U is the perpendicular bisector of segment IJ. Line L is the reflection of line U across line R. Line P passes through points H and Z. Point Q is the intersection of lines L and P. Find the length of segment BQ. 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 L and circumradius 12. Let K be the circumcircle of polygon ABCDEF. Point O is the midpoint of segment FL. Let H be the inversion of point O in circle K. Point R is the reflection of point L across point C. What is the distance between points R and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGIJK is a regular 10-gon with center H and circumradius 10. Line V is the angle bisector of angle GEI. Let U be the midpoint of segment BD. Let P be the line through point F perpendicular to line V. Point X is the reflection of point U across line P. Find the length of segment XE. 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 2. Point R is obtained by rotating point A by angle pi/12 about point G. Point W is obtained by rotating point B by angle pi/12 about point A. Let T be the circumcircle of polygon ABDEFG. Let O be the perpendicular bisector of segment CA. Line L passes through points R and E. Let J be the reflection of circle T across line L. Point H is the inversion of point C with respect to circle J. Point U is the reflection of point W across line O. What is the length of segment HU? 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 R and circumradius 9. Let U be the circumcircle of polygon ABCDE. Let M be the reflection of circle U through point A. Let J be the perpendicular bisector of segment RB. Let T be the reflection of circle M across line J. Point L is the midpoint of segment EA. Point S is the inversion of point A with respect to circle T. Find 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]
ABCDEFGH is a regular octagon with center Y and circumradius 5. Let O be the reflection of point C across point A. Let Q be the perpendicular bisector of segment OY. Line R is the reflection of line Q across point F. Let V be the line through point Y parallel to line R. Point M is the intersection of segment DB and line V. What is the distance between points O and M? 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 Z and circumradius 11. Line S is the angle bisector of angle ACB. Let M be the circumcircle of polygon ABCDEF. Let U be the reflection of circle M across line S. Let N be the inversion of point F in circle U. Find the length of segment NF. 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 S and circumradius 10. Line Q passes through point C parallel to segment FJ. Point Z is the reflection of point L across line Q. Point P is the midpoint of segment EZ. What is the distance between points K 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 R and circumradius 10. Line Y is the perpendicular bisector of segment CB. Let W be the line through point F parallel to line Y. Point U is the intersection of line W and segment AE. 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 ABCDE be a regular 5-gon with center X and circumradius 12. Let N be the line through points B and E. Let J be the line through point X parallel to line N. Point I is the midpoint of segment AC. Let U be the line through point D parallel to line J. Let H be the reflection of line J across line U. Point Y is the reflection of point I across line H. Find the length of segment EY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEF is a regular 5-gon with center D and circumradius 10. Point T is the reflection of point B across point D. Line N is the perpendicular bisector of segment TB. Point M is the reflection of point C across segment BA. Let Z be the line through point F perpendicular to line N. Point J is the reflection of point C across line Z. What is the measure of angle JAM, 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 L and circumradius 4. Let H be the circumcircle of polygon ABCDEF. Let P be the midpoint of segment DL. Point I is the inversion of point P with respect to circle H. Point V is the midpoint of segment CD. What is the distance between points I and V? 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 X and circumradius 5. Let S be the reflection of point K across point C. Let O be the angle bisector of angle SGE. Line T is the reflection of line O across point J. Point N is the reflection of point E across line T. Find the distance between points I and N. 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 3. Let N be the line through point E perpendicular to segment BE. Point T is the reflection of point G across segment BE. Point X is the reflection of point T across line N. Find the length of segment LX. 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 9. Let V be the reflection of point I across point B. Let T be the perpendicular bisector of segment EV. Let U be the angle bisector of angle CIV. Let Q be the reflection of line T through point I. Let Z be the reflection of line U through point C. Lines Q and Z intersect at point L. Find the distance between points L 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 ABCDEF be a regular hexagon with center M and circumradius 9. Point L is the reflection of point M across point A. Point G is the reflection of point B across point F. Let N be the midpoint of segment GL. Let X be the midpoint of segment EM. What is the distance between points N and X? 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. Let T be the perpendicular bisector of segment ID. Line M is the reflection of line T across point E. Let Y be the circumcircle of polygon ABCDEFG. Let Z be the reflection of circle Y through point C. Let N be the midpoint of segment IC. Point L is the reflection of point C across segment ND. Line M and segment ND intersect at point R. Let J be the inversion of point L in circle Z. 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 P and circumradius 7. Point Q is the reflection of point D across point C. Let T be the reflection of point Q across point E. Let I be the midpoint of segment CQ. Point V is the reflection of point P across segment IT. Find the distance between points V 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 S and circumradius 9. Let J be the angle bisector of angle CDE. Line O is the perpendicular bisector of segment EB. Line M is the reflection of line J across line O. Point U is the reflection of point B across point C. Let Q be the line through point E parallel to segment CA. Lines Q and M intersect at point R. What is the length of segment RU? 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 10. Let K be the line through points F and E. Let S be the line through point C parallel to line K. Let L be the reflection of line S across line K. Point T is the reflection of point F across line L. What is the measure of angle BFT, 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 ABCDEFGI be a regular octagon with center H and circumradius 6. Let X be the midpoint of segment IB. Let W be the circumcircle of polygon ABCDEFGI. Let P be the inversion of point X in circle W. Line Q is the perpendicular bisector of segment PE. Let J be the reflection of circle W through point P. Circle Z is the reflection of circle J across line Q. Point S is the inversion of point G with respect to circle Z. What is the length of segment DS? 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 N and circumradius 12. Line I is the perpendicular bisector of segment NE. Let S be the midpoint of segment NC. Let T be the rotation of point S by angle pi/3 around point C. Line U passes through points S and D. Point R is the midpoint of segment SD. Let L be the reflection of line U across line I. Let J be the angle bisector of angle SRT. Point Y is the reflection of point B across point E. Let P be the reflection of line J across line L. Point K is the reflection of point D across line P. What is the length of segment KY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFGHI be a regular octagon with center B and circumradius 10. Let X be the reflection of point F across point G. Let Z be the line through points H and X. Line S passes through point E perpendicular to line Z. Point K is the reflection of point F across line Z. Point N is the reflection of point K across line S. What is 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]
ABCDEFGI is a regular octagon with center H and circumradius 7. Point T is the reflection of point G across point I. Let L be the midpoint of segment BD. Point M is obtained by rotating point C by angle 3pi/4 about point D. Line O is the perpendicular bisector of segment MF. Point Q is the reflection of point T across line O. Let Z be the perpendicular bisector of segment EQ. Point P is the reflection of point M across line Z. 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]
ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 1. Line R is the angle bisector of angle DHI. Point W is the reflection of point F across line R. Line S passes through points I and E. Point X is the reflection of point L across line S. What is the distance between points X and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJLM is a regular dodecagon with center K and circumradius 7. Let N be the line through points A and K. Line R passes through point C parallel to line N. Line V is the reflection of line R across point H. Point T is the reflection of point L across line V. Find the distance between points T and I. 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 6. Line T passes through points B and F. Let V be the line through points A and D. Line Z passes through point D parallel to line T. Line L is the reflection of line Z across point B. Let M be the reflection of point D across point B. Point Y is the midpoint of segment DA. Point W is the midpoint of segment MY. Point G is the intersection of lines L and V. What is the distance between points G 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 S and circumradius 1. Let T be the midpoint of segment BS. Let O be the perpendicular bisector of segment AT. Point X is the reflection of point B across line O. Let V be the midpoint of segment DX. Let Q be the reflection of point B across point V. Point J is the midpoint of segment FE. What is the length of segment JQ? 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 K and circumradius 6. Line L is the angle bisector of angle FCG. Let X be the circumcircle of polygon ABCDEFGH. Let O be the reflection of circle X through point H. Let Q be the inversion of point A in circle O. Point U is the reflection of point E across line L. Find the distance between points Q 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 ABCDE be a regular 5-gon with center R and circumradius 3. Let W be the reflection of point B across point R. Let K be the circumcircle of polygon ABCDE. Circle Y is the reflection of circle K across point C. Point Q is the inversion of point B with respect to circle Y. Let I be the perpendicular bisector of segment EW. Point J is the reflection of point B across line I. Find the distance between points J 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 ABCDEFGHIJ be a regular 10-gon with center S and circumradius 3. Let Q be the midpoint of segment BA. Point Y is the reflection of point G across point I. Let O be the line through point G parallel to segment BA. Point K is the reflection of point Y across line O. Find 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]
ABCDEFGHIJ is a regular 10-gon with center R and circumradius 2. Let K be the rotation of point E by angle 2pi/3 around point G. Point Z is the midpoint of segment IB. Let P be the line through points E and Z. Line Y passes through point G parallel to line P. Point N is the reflection of point K across line Y. Find the distance between points N and K. 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 1. Point T is the midpoint of segment JG. Let R be the midpoint of segment KT. Line V passes through point L parallel to segment GJ. Point N is the intersection of line V and segment RD. Find the measure of angle KHN, 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 ABCDEFGHIJK be a regular 11-gon with center Z and circumradius 9. Point P is the reflection of point K across point I. Let X be the midpoint of segment ZK. Let L be the line through points X and P. Point U is the intersection of line L and segment JZ. What is the length of segment GU? 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 7. Point N is the reflection of point H across point D. Line L is the angle bisector of angle ADC. Line O passes through point N parallel to line L. Point T is the reflection of point I across line O. What is the length of segment DT? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 7. Point Y is the reflection of point G across segment DI. Let V be the line through points A and G. Let X be the line through point Y parallel to segment EH. Point U is the reflection of point E across point M. Point W is the intersection of lines V and X. What is the length of segment WU? 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. Point S is the midpoint of segment AF. Let I be the midpoint of segment CD. Let G be the midpoint of segment AS. Point N is the midpoint of segment IO. Find the distance between points N and G. 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 2. Line N passes through points I and H. Line P is the perpendicular bisector of segment AC. Let S be the reflection of line P across line N. Lines S and N intersect at point J. What is the distance between points J 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 ABCDEFHIJK be a regular 10-gon with center G and circumradius 2. Let Z be the rotation of point B by angle pi/12 around point J. Line U is the angle bisector of angle EIK. Line X passes through point B parallel to line U. Point N lies on line X. Let S be the line through point N parallel to segment GZ. Let Q be the line through point A parallel to line S. Point L is the reflection of point B across line Q. What is the length of segment LD? 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. Let T be the perpendicular bisector of segment BA. Let U be the line through points D and J. Line M is the reflection of line T across line U. Point O is the reflection of point G across line M. Find the length of segment JO. 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 X and circumradius 3. Point M is the midpoint of segment CD. Point I is the midpoint of segment GM. Let N be the reflection of point C across point I. What is the length of segment EN? 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 3. Point X is the reflection of point C across point B. Let O be the perpendicular bisector of segment DL. Line K passes through point L perpendicular to line O. Point P is the midpoint of segment AC. Point J is the reflection of point X across line K. Find the distance between points J and P. 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 L and circumradius 9. Let V be the line through points H and A. Point Q is the reflection of point J across line V. Point P is the reflection of point C across line V. Point R is the reflection of point L across line V. Point M is the reflection of point R across point Q. Find the distance between points M 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 7. Let K be the midpoint of segment CB. Let P be the midpoint of segment CN. Let U be the reflection of point K across point B. Let W be the circumcircle of polygon ABCDEFGH. Let T be the inversion of point P in circle W. Let R be the reflection of point U across point A. Find 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]
ABCDEFGH is a regular octagon with center K and circumradius 8. Line P passes through points F and A. Point Y is the reflection of point H across line P. Let M be the midpoint of segment YK. What is the length of segment CM? 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 N and circumradius 2. Let T be the circumcircle of polygon ABCDEFG. Point S is the reflection of point E across point D. Let X be the midpoint of segment SD. Point R is the midpoint of segment CA. Let K be the inversion of point R in circle T. 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]
JY = 2. ABCDEFGH is a regular octagon with center Y and circumradius 3. Let I be the circumcircle of polygon ABCDEFGH. Let T be the reflection of circle I through point F. Let O be the inversion of point G in circle T. Let S be the midpoint of segment AH. Find the distance between points O 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 ABCDEFGI be a regular octagon with center H and circumradius 12. Point W is the midpoint of segment CI. Let Q be the reflection of point H across point W. Let Z be the angle bisector of angle WEQ. Point K is the reflection of point A across line Z. Find the distance between points K 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 ACDEFGHI be a regular octagon with center B and circumradius 3. Line Z passes through points B and A. Let O be the circumcircle of polygon ACDEFGHI. Line T passes through point I parallel to line Z. Let P be the line through point H parallel to line T. Circle R is the reflection of circle O across line P. Point W is the inversion of point F with respect to circle R. What is the distance between points G and W? 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 R and circumradius 10. Let Q be the angle bisector of angle IFH. Let V be the circumcircle of polygon ABCDEFGHIJKL. Let Y be the reflection of circle V across line Q. Let O be the inversion of point R in circle Y. Find the length of segment OC. 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. Point X is obtained by rotating point C by angle 7pi/12 about point F. Point W is the reflection of point A across point X. Let L be the reflection of point W across point B. Let J be the circumcircle of polygon ABCDF. Let P be the reflection of point L across point X. Let Z be the inversion of point P in circle J. Point K is the reflection of point F across point Z. Find the distance between points K 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 ABCDEFGHIJK be a regular 11-gon with center M and circumradius 2. Line P is the angle bisector of angle DCK. Let U be the midpoint of segment FK. Let Q be the reflection of point A across point H. Point X is the reflection of point Q across line P. What is the distance between points U 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 ABCDFGHI be a regular octagon with center E and circumradius 6. Point T is the reflection of point B across point D. Point V is obtained by rotating point T by angle 5pi/6 about point A. Let W be the rotation of point V by angle 3pi/4 around point G. Point L is the reflection of point E across segment GW. Let P be the line through points L and G. Point R is the reflection of point C across line P. Let Y be the reflection of point C across point R. Find the length of segment HY. 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 G and circumradius 11. Let M be the rotation of point B by angle pi/12 around point C. Line Z passes through point C perpendicular to segment CB. Line L passes through point D parallel to line Z. Let X be the midpoint of segment ME. Point S is the reflection of point X across line L. Find the length of segment SA. 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 L and circumradius 11. Point P is the midpoint of segment EL. Let W be the midpoint of segment DB. Point R is the midpoint of segment PA. Let X be the angle bisector of angle HFG. Point M is the reflection of point W across line X. What is the distance between points R and M? 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 9. Let Z be the angle bisector of angle MGH. Point T is the reflection of point L across line Z. Point U is the reflection of point T across segment ME. Let V be the midpoint of segment LC. Let P be the midpoint of segment UV. Find 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]
Let ABCDE be a regular 5-gon with center Q and circumradius 7. Point H is obtained by rotating point B by angle pi about point A. Let P be the midpoint of segment DB. Point J is the midpoint of segment BP. Let Z be the perpendicular bisector of segment CH. Point N is the reflection of point J across line Z. What is the distance between points E and N? 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 V and circumradius 1. Let S be the angle bisector of angle BFV. Let U be the line through point H parallel to line S. Line P passes through point D parallel to line U. Point Q is the reflection of point F across line P. What is the distance between points B 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 ABCDFGHIJKLM be a regular dodecagon with center E and circumradius 3. Let O be the reflection of point H across point I. Let X be the midpoint of segment LK. Point Y is the reflection of point X across point O. Line V is the perpendicular bisector of segment LK. Point S is the reflection of point Y across line V. Find the length of segment AS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFGHIJ is a regular 9-gon with center A and circumradius 1. Let V be the midpoint of segment IG. Line X passes through point D parallel to segment FA. Point M is the reflection of point V across line X. Find the length of segment CM. 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 J and circumradius 8. Let U be the midpoint of segment AB. Point Z is the midpoint of segment UC. Line K is the angle bisector of angle EDA. Point O is the reflection of point U across line K. Point P is the reflection of point O across segment BZ. Find the distance between points P 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 ABCDEFGI be a regular octagon with center H and circumradius 9. Line L passes through points I and E. Point W is the reflection of point F across line L. Point P is the midpoint of segment WB. Find the length of segment PI. 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 8. Let O be the line through point B perpendicular to segment AB. Point H is the reflection of point V across line O. Point X is on segment AB. Find the measure of angle XBH, 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 ABCE be a square with center D and circumradius 7. Let I be the circumcircle of polygon ABCE. Line Z is the perpendicular bisector of segment BE. Line R passes through point B parallel to line Z. Let H be the reflection of circle I across line R. Point J is the inversion of point C with respect to circle H. Point M is the reflection of point D across segment JA. Let Y be the inversion of point A in circle H. Point Q is the reflection of point M across segment YJ. Find the length of segment QJ. 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 11. Let Q be the reflection of point E across point D. Line P is the perpendicular bisector of segment FI. Line T passes through point J parallel to line P. Point V is the reflection of point L across line T. Let X be the midpoint of segment GV. Find the length of segment QX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 7. Let Y be the line through points K and H. Line U passes through point D parallel to line Y. Let Z be the midpoint of segment GE. Let P be the reflection of line U through point Z. Let N be the circumcircle of polygon ACDEFGHIJKLM. Circle T is the reflection of circle N across line P. Point O is the inversion of point Z with respect to circle T. What is the length of segment OB? 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 2. Line Y is the perpendicular bisector of segment KC. Let X be the line through point C parallel to line Y. Let P be the line through points E and J. Line Z is the reflection of line P across line X. Point Q is the reflection of point K across line Z. Let T be the midpoint of segment QH. What is the measure of angle EMT, 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 ABDEFG be a regular hexagon with center C and circumradius 9. Point R is the reflection of point F across point E. Let H be the reflection of point R across point C. Let X be the perpendicular bisector of segment HC. Let Z be the line through point E parallel to line X. Point N is the intersection of segment RA and line Z. Find the distance between points G and N. 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 Y and circumradius 3. Line Q is the angle bisector of angle HGA. Let M be the line through point F perpendicular to line Q. Point T is the reflection of point Y across line M. Point R is the reflection of point T across line Q. Find the length of segment RY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGIJKL is a regular 11-gon with center H and circumradius 8. Let R be the angle bisector of angle CIA. Line V is the reflection of line R across point J. Let N be the midpoint of segment GL. Line X passes through point F parallel to line V. Let Y be the midpoint of segment GF. Let M be the midpoint of segment CY. Point P is the reflection of point M across line X. What is the distance between points P and N? 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 L and circumradius 9. Line O is the perpendicular bisector of segment GL. Let M be the line through point F parallel to line O. Line Y is the reflection of line O across line M. Point S is the reflection of point A across line Y. Find the length of segment FS. 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. Line Y is the angle bisector of angle BPE. Let Z be the line through point A parallel to line Y. Point L is the midpoint of segment EP. Point H is the intersection of segment BL and line Z. What is the distance between points H and B? 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 1. Point I is the reflection of point C across point G. Let M be the perpendicular bisector of segment XD. Point Q is the reflection of point F across line M. Let N be the midpoint of segment EI. Point S is the midpoint of segment NQ. Find the length of segment SI. 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 O and circumradius 1. Let S be the circumcircle of polygon ABCDEFGHIJKL. Let Q be the reflection of circle S through point A. Line N is the perpendicular bisector of segment IH. Let W be the inversion of point F in circle Q. Let Z be the line through point W perpendicular to line N. Point X is the reflection of point L across line Z. What is the length of segment XJ? 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 V and circumradius 2. Let Z be the angle bisector of angle EDC. Line G is the angle bisector of angle ECV. Point M is the reflection of point B across line G. Point K is the reflection of point M across line Z. Point L is the reflection of point M across point A. Point U is the reflection of point L across point K. What is the length of segment MU? 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 N and circumradius 8. Let S be the line through points G and I. Line Q passes through point L parallel to line S. Segment NB and line Q intersect at point M. Find the distance between points M and L. 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 L passes through points B and A. Point K is the reflection of point E across line L. Point M is the reflection of point K across point B. Point T is the reflection of point C across point M. Find the length of segment KT. 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 P and circumradius 6. Let L be the perpendicular bisector of segment PB. Point R is the reflection of point A across line L. Point G is the midpoint of segment AR. What is the length of segment BG? 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 K and circumradius 4. Let W be the rotation of point B by angle 2pi/3 around point H. Point J is obtained by rotating point W by angle pi/6 about point A. Line S is the perpendicular bisector of segment JG. Point M is the reflection of point B across line S. Point Z is the midpoint of segment HA. Find the length of segment MZ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDFG is a regular hexagon with center E and circumradius 5. Point T is the midpoint of segment AB. Point O is the reflection of point T across point C. Let S be the midpoint of segment EC. 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]
ABCDEGHIJK is a regular 10-gon with center F and circumradius 5. Let S be the line through point D perpendicular to segment GB. Line O is the reflection of line S across point G. Let Y be the angle bisector of angle AEI. Line N is the reflection of line Y across line O. Point L is the reflection of point I across line N. What is the length of segment LF? 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 10. Line R is the angle bisector of angle HFD. Let L be the line through point A parallel to line R. Line X is the reflection of line L across line R. Let T be the midpoint of segment ZD. Point Y is the reflection of point T across line X. What is the distance between points B and Y? 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 L and circumradius 12. Line X is the perpendicular bisector of segment BI. Line J passes through point B parallel to line X. Let S be the reflection of line J through point D. Point V is the reflection of point C across line S. Let R be the reflection of point V across point E. What is the length of segment IR? 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 I and circumradius 6. Let O be the perpendicular bisector of segment FE. Line P passes through point B parallel to line O. Let Y be the line through point E parallel to line P. Point V is the reflection of point F across line Y. Find the length of segment AV. 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 6. Let M be the midpoint of segment CF. Let N be the midpoint of segment FM. Let T be the midpoint of segment BP. Find 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]
ABCDEFHIJKLM is a regular dodecagon with center G and circumradius 9. Point R is the reflection of point C across point D. Let X be the line through points K and R. Line Z passes through point L parallel to segment FG. Point V is the intersection of lines X and Z. Find the distance between points V and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFGHI be a regular octagon with center B and circumradius 2. Line W is the angle bisector of angle HEA. Let O be the midpoint of segment CI. Line Q is the reflection of line W across point A. Point P is the reflection of point G across line Q. Find the length of segment OP. 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 W and circumradius 8. Let L be the line through points W and C. Let K be the line through point F parallel to line L. Let M be the reflection of line K through point D. Point R is the reflection of point H across line M. Find the length of segment RW. 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 G and circumradius 5. Let Y be the line through points A and E. Let L be the perpendicular bisector of segment ED. Let M be the perpendicular bisector of segment EC. Let V be the rotation of point B by angle pi/3 around point E. Line O is the reflection of line Y across point V. Let K be the reflection of line L across line M. Lines O and K intersect at point I. Let R be the reflection of point C across point I. What is the length of segment GR? 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 4. Let L be the line through point G perpendicular to segment DE. Point K is the reflection of point A across line L. Point S is the midpoint of segment FK. What is the length of segment SB? 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 H and circumradius 7. Let Q be the line through points C and B. Line R is the angle bisector of angle FBA. Line L is the reflection of line Q across line R. Point Z is the reflection of point E across line L. Find the length of segment DZ. 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 1. Let N be the circumcircle of polygon ABCDE. Let V be the reflection of point J across point C. Let K be the inversion of point V in circle N. Let H be the midpoint of segment JK. What is the measure of angle DCH, in radians? 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 Q and circumradius 3. Let Y be the angle bisector of angle FEA. Let Z be the reflection of line Y through point B. Point X is the reflection of point L across line Z. Point T is the midpoint of segment XA. What is the length of segment TD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEF is a regular 5-gon with center D and circumradius 4. Let W be the perpendicular bisector of segment BF. Line S passes through point D perpendicular to line W. Line R is the reflection of line W across point C. Let M be the line through point F perpendicular to line R. Let I be the reflection of line S across line M. Point V is the reflection of point F across line I. Point N is the reflection of point C across point F. What is the length of segment VN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BCDEF be a regular 5-gon with center A and circumradius 1. Point T is the midpoint of segment FD. Let P be the reflection of point F across point C. Let S be the angle bisector of angle PAB. Point U is the reflection of point F across line S. Point V is the reflection of point U across point T. Find the distance between points V 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 ABCDEFHI be a regular octagon with center G and circumradius 1. Let J be the line through point H perpendicular to segment AE. Let Y be the reflection of line J through point D. Point T is the reflection of point H across line Y. What is the length of segment IT? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFGHIJ be a regular 9-gon with center B and circumradius 11. Let L be the midpoint of segment IG. Point X is the reflection of point A across segment IG. Point K is the reflection of point L across segment HX. What is the distance between points K and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEFG be a regular hexagon with center D and circumradius 6. Point R is the midpoint of segment CE. Let U be the circumcircle of polygon ABCEFG. Line J passes through points A and B. Let Z be the inversion of point R in circle U. Line K is the reflection of line J across point Z. Point M is the reflection of point A across line K. What is the distance between points M and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGIJK is a regular 10-gon with center H and circumradius 5. Point V is the midpoint of segment JE. Let W be the midpoint of segment FK. Point O is the midpoint of segment IW. 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]
ABCDEFGHJKL is a regular 11-gon with center I and circumradius 8. Let Z be the perpendicular bisector of segment IE. Let S be the reflection of point J across point H. Point Y is the reflection of point C across segment BA. Let U be the circumcircle of polygon ABCDEFGHJKL. Let T be the reflection of circle U across line Z. Point O is the inversion of point S with respect to circle T. Find the distance between points Y 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 ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 5. Let O be the midpoint of segment JG. Let P be the line through point O perpendicular to segment KH. Point T is the reflection of point D across line P. Find the length of segment TD. 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 reflection of point E across point A. Let G be the rotation of point W by angle 5pi/6 around point D. Let N be the reflection of point B across point C. Let J be the midpoint of segment NE. Let H be the reflection of point D across point C. Point P is the midpoint of segment HG. Line Z is the angle bisector of angle CPW. Point I is the reflection of point H across line Z. Find the distance between points J and I. 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 X and circumradius 7. Point K is the reflection of point J across point H. Point U is obtained by rotating point E by angle 5pi/6 about point I. Let T be the reflection of point B across point D. Point M is the midpoint of segment ET. Let Z be the line through points T and H. Let R be the line through points K and X. Let L be the line through point U parallel to line Z. Point S is the intersection of lines R and L. Point W is the reflection of point S across segment AB. Find the distance between points M and W. 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 R and circumradius 6. Point Y is the reflection of point R across point D. Let V be the line through points E and A. Let P be the line through points D and C. Point X is the intersection of lines P and V. Point U is the midpoint of segment YX. What is the distance between points U 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 6. Point K is the midpoint of segment AC. Point P is the midpoint of segment ZC. Let X be the circumcircle of polygon ABCDE. Let N be the inversion of point P in circle X. Find the measure of angle NZK, 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 J and circumradius 6. Let K be the rotation of point C by angle 5pi/12 around point J. Let P be the reflection of point D across point B. Let Q be the midpoint of segment CK. Let G be the midpoint of segment PK. 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]
Let ABCDEFGHI be a regular 9-gon with center Q and circumradius 7. Point J is the reflection of point Q across point B. Let W be the midpoint of segment JF. Segments EH and FD intersect at point O. What is the distance between points O and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDE is a square with center A and circumradius 2. Point J is obtained by rotating point E by angle pi/12 about point B. Let L be the line through point J perpendicular to segment CB. Point T is the reflection of point D across line L. Point U is the reflection of point T across point E. Let I be the midpoint of segment BE. What is the length of segment IU? 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 12. Point O is the midpoint of segment AF. Let W be the midpoint of segment AE. Let X be the reflection of point W across point B. Point L is the reflection of point X across point D. Let I be the midpoint of segment WO. Find the length of segment LI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFGH be a regular 7-gon with center C and circumradius 12. Let I be the reflection of point B across point C. Let Q be the midpoint of segment IA. Let S be the line through points A and H. Point M is the reflection of point Q across line S. Find the distance between points M 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 ABCDEFGH be a regular octagon with center V and circumradius 8. Let W be the midpoint of segment FH. Let L be the circumcircle of polygon ABCDEFGH. Circle X is the reflection of circle L across point W. Let P be the inversion of point E in circle X. Find the distance between points D 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 ABCDEF be a regular hexagon with center P and circumradius 11. Let U be the rotation of point P by angle pi around point F. Line T is the angle bisector of angle DBU. Let S be the line through points A and C. Let X be the reflection of line S through point U. Let V be the line through point D perpendicular to line S. Let L be the reflection of line V across line T. Lines L and X intersect at point H. What is the length of segment AH? 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 K and circumradius 11. Let T be the angle bisector of angle CAF. Point P is on line T. Let N be the line through points C and P. Point U is the reflection of point B across line N. Find the distance between points C and U. 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 6. Let Z be the midpoint of segment FJ. Line P is the angle bisector of angle KZE. Let V be the line through point D parallel to line P. Line R passes through point B perpendicular to line V. Point U is the reflection of point L across line R. What is the distance between points C and U? 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 W and circumradius 12. Line Q passes through points D and B. Line T is the reflection of line Q across point A. Let O be the line through points C and E. Point U is the intersection of lines O and T. Line H passes through point A perpendicular to line Q. Point Z is on line H. What is the measure of angle UAZ, in radians? 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. Line W passes through points C and F. Point R lies on line W. Let U be the angle bisector of angle ESR. Point Z is the reflection of point L across line U. What is the distance between points Z 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 ABCDEFGHI be a regular 9-gon with center O and circumradius 7. Point M is the reflection of point A across segment HF. Let Q be the reflection of point A across point I. Let Z be the reflection of point M across point H. What is the distance between points Z 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 ABCDEF be a regular hexagon with center T and circumradius 11. Let W be the rotation of point E by angle pi/12 around point D. Line V is the perpendicular bisector of segment BW. Let Z be the angle bisector of angle CAT. Let X be the circumcircle of polygon ABCDEF. Point G is the reflection of point F across line V. Let S be the reflection of point B across point G. Line K is the reflection of line Z across point C. Let L be the reflection of circle X through point F. Line H is the reflection of line K across line V. Let R be the line through point W parallel to line H. Let M be the reflection of circle L across line R. Point J is the inversion of point S with respect to circle M. Find the distance between points S 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 ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 7. Line O passes through points L and K. Point Q is obtained by rotating point F by angle pi/4 about point K. Point T is the midpoint of segment AQ. Line P passes through point T perpendicular to line O. Let X be the reflection of line P through point C. Point M is the reflection of point K across line X. Find the length of segment MA. 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 O and circumradius 6. Let F be the line through points A and D. Let E be the line through point C perpendicular to line F. Point T is the reflection of point O across line E. Line Q passes through point T perpendicular to line E. Let J be the midpoint of segment TB. Let W be the reflection of line Q through point A. Let G be the reflection of point J across point O. Point R is the reflection of point G across line W. 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]
ABCDEFGHIJKL is a regular dodecagon with center V and circumradius 4. Line R passes through point G perpendicular to segment KL. Point Z is the reflection of point E across line R. Line Y passes through points A and G. Let O be the line through points E and Z. Point Q is the intersection of lines O and Y. Find the distance between points Q and J. 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 Z and circumradius 5. Let W be the circumcircle of polygon ABCDEFGHIJKL. Let M be the reflection of point K across point L. Let Y be the perpendicular bisector of segment ML. Let O be the reflection of circle W across line Y. Point V is the inversion of point L with respect to circle O. What is the length of segment VL? 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. Line K is the angle bisector of angle EAC. Point Y is the midpoint of segment BD. Let Q be the midpoint of segment YD. Let O be the angle bisector of angle EQY. Point J is the reflection of point B across line K. Point W is the reflection of point J across point Z. Point F is the reflection of point Z across line O. Point N is the reflection of point F across point W. What is the distance between points D and N? 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 7. Point T is the midpoint of segment GK. Let Y be the midpoint of segment DT. Find the length of segment IY. 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 S and circumradius 2. Let O be the perpendicular bisector of segment BJ. Let W be the circumcircle of polygon ABCDEFGHIJK. Let Z be the line through point F perpendicular to line O. Let T be the reflection of circle W across line Z. Let Q be the inversion of point G in circle T. Find the distance between points H and Q. 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 Q and circumradius 4. Let V be the circumcircle of polygon ABCDEFGHIJKL. Let M be the reflection of circle V through point B. Point S is the midpoint of segment BJ. Let P be the inversion of point E in circle M. 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 BCDEFG be a regular hexagon with center A and circumradius 12. Point V is the reflection of point F across point G. Let Z be the circumcircle of polygon BCDEFG. Circle K is the reflection of circle Z across point F. Let S be the reflection of circle K through point A. Line I is the angle bisector of angle AVE. Let O be the reflection of circle S through point B. Point M is the reflection of point E across line I. Point H is the inversion of point F with respect to circle O. What is the distance between points H and M? Express your answer as a decimal number rounded to 2 decimal places.

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