[User]
ABCDEFGHI is a regular 9-gon with center X and circumradius 9. Let W be the line through points B and E. Line M is the angle bisector of angle BHD. Line Q passes through point A parallel to line W. Point V is the intersection of line Q and segment DG. Let O be the line through point F perpendicular to line M. Point S is the reflection of point V across line O. What is the measure of angle SFE, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center M and circumradius 8. Point F is the midpoint of segment EC. Point U is the reflection of point F across point B. Let L be the midpoint of segment UA. Find the length of segment LC. 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 9. Point Y is the midpoint of segment FH. Let T be the angle bisector of angle IKJ. Line P passes through point A perpendicular to line T. Point S is the reflection of point I across line P. What is the length of segment SY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 9. Point W is the midpoint of segment AI. Let P be the reflection of point E across point G. Point O is the reflection of point B across point W. What is the distance between points O and P? 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 8. Let Q be the reflection of point G across point A. Let V be the midpoint of segment DQ. Let U be the reflection of point Z across point Q. Find the length of segment VU. 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 3. Let V be the perpendicular bisector of segment FC. Line T is the reflection of line V across point L. Point M is the reflection of point L across line T. Line N passes through point M parallel to line T. Point R is the reflection of point S across line N. Find the length of segment RD. 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 7. Let X be the rotation of point I by angle pi/4 around point D. Let W be the line through point Z parallel to segment DB. Let S be the line through point D perpendicular to line W. Point N is the reflection of point X across line S. What is the distance between points N and A? 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 5. Point U is the reflection of point O across point J. Line P passes through point D perpendicular to segment HE. Line Q passes through point U parallel to segment HE. Line N passes through point G parallel to line P. Point Y is the intersection of lines N and Q. Let X be the midpoint of segment DY. What is the distance between points B and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center Y and circumradius 1. Line P passes through point G perpendicular to segment LH. Let O be the reflection of line P through point L. Let Z be the reflection of line O across line P. Point W is the reflection of point A across line Z. What is the distance between points W and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center X and circumradius 1. Let T be the angle bisector of angle XDC. Let L be the line through point A perpendicular to line T. Line G is the reflection of line L across point E. Let W be the line through point E perpendicular to line G. Point F is the reflection of point C across line W. What is the length of segment BF? 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 12. Let N be the line through points C and X. Let P be the line through point B parallel to line N. Let S be the reflection of point I across point C. Point Q is the intersection of line P and segment JG. Find the length of segment SQ. 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 Y and circumradius 8. Line T is the angle bisector of angle YAJ. Point N is the reflection of point L across line T. Let M be the reflection of point G across point H. Point Q is the reflection of point M across line T. What is the distance between points Q and N? 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 2. Point I is the reflection of point F across point B. Let J be the rotation of point E by angle pi/2 around point C. Line S is the angle bisector of angle BDJ. Let W be the midpoint of segment FA. Point T is the reflection of point W across line S. Let R be the reflection of point I across point W. What is the length of segment RT? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center Y and circumradius 10. Let M be the rotation of point D by angle 3pi/4 around point A. Let N be the circumcircle of polygon ABCDEF. Let S be the line through point M perpendicular to segment DA. Let I be the reflection of circle N across line S. Point K is the inversion of point F with respect to circle I. What is the length of segment KE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDFG be a regular hexagon with center E and circumradius 4. Let Z be the rotation of point B by angle pi/4 around point E. Point I is the midpoint of segment FC. Let T be the line through points E and Z. Point X is on line T. Line L passes through points I and X. Point R is the reflection of point Z across line L. Find 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 P and circumradius 12. Point J is the midpoint of segment HF. Let L be the midpoint of segment IJ. Point W is the reflection of point L across segment GB. What is the length of segment WP? 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 5. Line T is the angle bisector of angle CEF. Let P be the midpoint of segment HI. Let M be the angle bisector of angle IBE. Point X is the reflection of point P across line M. Let N be the midpoint of segment PG. Let J be the line through point N parallel to line T. Point Z is the reflection of point X across line J. Point Q is the midpoint of segment ZH. What is the length of segment QH? 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 K and circumradius 6. Let Q be the line through point K parallel to segment EF. Line T passes through points B and D. Lines T and Q intersect at point M. Point U is the reflection of point D across segment EF. Find the length of segment UM. 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 Z and circumradius 2. Let U be the perpendicular bisector of segment ZC. Let S be the line through point A parallel to line U. Segment DH and line S intersect at point M. Point T is the reflection of point Z across segment DH. What is the distance between points M and T? 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 3. Let N be the reflection of point F across point G. Let O be the line through points N and E. Let Q be the line through point H parallel to line O. Point R is the reflection of point C across line Q. Let S be the midpoint of segment BA. Find the length of segment SR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center V and circumradius 3. Let K be the reflection of point C across point G. Point N is the reflection of point F across point K. Let L be the midpoint of segment HA. Line S passes through points L and N. Point I is the reflection of point E across line S. What is the distance between points I and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center P and circumradius 7. Let X be the circumcircle of polygon ABCDEFGHIJKL. Let V be the reflection of circle X through point G. Let T be the reflection of point C across point J. Let U be the reflection of circle V through point C. Point M is the inversion of point T with respect to circle U. What is the distance between points J 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 ABCDFGHIJK be a regular 10-gon with center E and circumradius 10. Line L passes through points K and G. Let Z be the line through point F parallel to line L. Point W is the reflection of point B across line Z. Let S be the midpoint of segment ID. What is the length of segment WS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ACDEFGHIJKLM is a regular dodecagon with center B and circumradius 11. Let Z be the circumcircle of polygon ACDEFGHIJKLM. Point N is the midpoint of segment AJ. Point P is the inversion of point N with respect to circle Z. What is the distance between points L and P? 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 6. Let Z be the rotation of point B by angle pi around point C. Let T be the perpendicular bisector of segment BA. Point V is the reflection of point Z across line T. Point N is the reflection of point V across segment BE. Point J is the midpoint of segment CN. What is the distance between points J and D? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEGHIJKL is a regular 11-gon with center F and circumradius 12. Let V be the circumcircle of polygon ABCDEGHIJKL. Let W be the midpoint of segment AE. Let P be the reflection of circle V through point A. Point Z is the reflection of point F across segment AE. Point S is the inversion of point W with respect to circle P. Find 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]
ABCDEFGH is a regular octagon with center L and circumradius 3. Let Y be the line through point B perpendicular to segment GH. Let P be the line through point E parallel to line Y. Point X is the reflection of point G across line P. Point J is the reflection of point D across segment GH. Find 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 Y and circumradius 3. Line G is the angle bisector of angle ACY. Let Z be the reflection of point E across point B. Let V be the line through point Z parallel to line G. Line H passes through point D parallel to line V. Point X is the reflection of point B across line H. Find the distance between points X 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 ABCDEFGH be a regular octagon with center Y and circumradius 5. Let U be the circumcircle of polygon ABCDEFGH. Let Z be the reflection of circle U through point A. Let O be the perpendicular bisector of segment HE. Point V is the midpoint of segment HA. Point S is the inversion of point V with respect to circle Z. Point K is on line O. Point J lies on line O at distance 1 from point K. Line T passes through point S parallel to line O. Let Q be the reflection of line T through point J. Let P be the midpoint of segment AB. Point M is the midpoint of segment EB. Let L be the line through point M parallel to line Q. Point I is the reflection of point P across line L. Find the length of segment IE. 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 Z and circumradius 4. Let J be the angle bisector of angle HEZ. Point S is the reflection of point D across line J. Let W be the circumcircle of polygon ABCDEFGH. Let U be the inversion of point S in circle W. 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]
ABCDEF is a regular hexagon with center L and circumradius 5. Let H be the circumcircle of polygon ABCDEF. Point O is obtained by rotating point B by angle pi/12 about point F. Line R is the perpendicular bisector of segment OB. Point U is the reflection of point L across line R. Let V be the line through points A and O. Let Z be the reflection of circle H across line V. Let Q be the inversion of point D in circle Z. Find the distance between points U and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center Q and circumradius 1. Let P be the perpendicular bisector of segment AB. Let L be the line through point E parallel to line P. Let U be the reflection of line L across line P. Let S be the line through point D perpendicular to segment BQ. Point Y is the reflection of point C across point E. Let J be the reflection of line P across line S. Let M be the line through point F parallel to line J. Line R passes through point Y parallel to line M. Lines R and U intersect at point I. Point O is the reflection of point I across line P. What is the distance between points A and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDFGHIJK be a regular 10-gon with center E and circumradius 9. Let L be the angle bisector of angle DFJ. Line X is the reflection of line L across point B. Point V is the reflection of point J across line X. Let W be the reflection of point V across point A. Find the length of segment WG. 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 Y and circumradius 5. Line K passes through points A and J. Point N is obtained by rotating point E by angle pi/12 about point H. Line Z is the angle bisector of angle BFN. Line M is the reflection of line K across line Z. Line P passes through point N parallel to line K. Point L is the intersection of lines P and M. Find the distance between points L and D. 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 1. Let M be the line through point J perpendicular to segment IC. Line Z is the reflection of line M across point L. Let Y be the line through point G parallel to line Z. Line Y and segment IC intersect at point R. What is the distance between points R 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 ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 9. Line P is the perpendicular bisector of segment JE. Let U be the perpendicular bisector of segment LB. Let T be the reflection of line U across line P. Line W is the reflection of line P across line T. Point X is the intersection of lines W and T. What is the length of segment EX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center I and circumradius 6. Line V is the perpendicular bisector of segment EF. Let O be the reflection of line V through point H. Line M passes through point C perpendicular to line O. Point Q is the reflection of point E across line M. Find the length of segment QB. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDFGHIJKLM be a regular dodecagon with center E and circumradius 1. Let Z be the reflection of point F across point D. Let V be the perpendicular bisector of segment LZ. Let P be the line through point G perpendicular to line V. Point R is the reflection of point M across line P. Point N is the midpoint of segment AB. Find the distance between points R and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center N and circumradius 12. Line O is the angle bisector of angle CFE. Let T be the line through point A perpendicular to line O. Let M be the circumcircle of polygon ABCDEF. Let R be the reflection of circle M across line T. Let V be the inversion of point C in circle R. 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 ABCD be a square with center G and circumradius 2. Line Q is the perpendicular bisector of segment DC. Let F be the line through point D perpendicular to line Q. Point H is the reflection of point A across line F. Point E is the reflection of point H across point G. Let L be the midpoint of segment CD. What is the length of segment EL? 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 4. Line X passes through points G and E. Let Y be the reflection of line X through point B. Let Z be the reflection of line Y through point H. Point T is the reflection of point B across line Z. 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 ABCDEFGHJKLM be a regular dodecagon with center I and circumradius 12. Let X be the line through points F and H. Line V passes through point E parallel to line X. Point Y is the intersection of line V and segment AG. What is the length of segment DY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCE is a square with center D and circumradius 12. Let F be the midpoint of segment AE. Let H be the reflection of point D across point F. Line R passes through point A perpendicular to segment CH. Point L is the reflection of point F across line R. Find the measure of angle LFB, 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 Q and circumradius 4. Line O passes through points K and J. Point U is the reflection of point F across line O. Let P be the midpoint of segment DK. 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]
ABCDEFGHIJKL is a regular dodecagon with center P and circumradius 7. Let R be the circumcircle of polygon ABCDEFGHIJKL. Let Q be the midpoint of segment IK. Point M is the inversion of point Q with respect to circle R. Let T be the midpoint of segment MH. Find the distance between points I and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center M and circumradius 1. Let W be the midpoint of segment AM. Line O is the angle bisector of angle DEB. Line Q is the reflection of line O across point A. Point Y is the reflection of point B across line Q. Let T be the midpoint of segment YE. Let X be the reflection of point W across point M. What is the distance between points T and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEG be a regular hexagon with center F and circumradius 6. Let K be the angle bisector of angle GFD. Point Z is the reflection of point F across point G. Point M is the midpoint of segment ZE. Point P is the reflection of point Z across line K. What is the length of segment PM? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDFG be a regular hexagon with center E and circumradius 3. Point S is the reflection of point D across point B. Let P be the circumcircle of polygon ABCDFG. Let T be the reflection of circle P through point S. Let Y be the reflection of circle T through point E. Point J is the inversion of point A with respect to circle Y. What is the length of segment JF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ACDEFGHIJKL is a regular 11-gon with center B and circumradius 7. Line V is the angle bisector of angle HBK. Line W passes through point C parallel to segment AK. Line M is the reflection of line V across line W. Point P is the intersection of segment AK and line M. Find the length of segment PB. 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 M and circumradius 4. Let Y be the angle bisector of angle BMG. Line Z passes through points F and H. Line K is the reflection of line Y across line Z. Point N is the reflection of point M across line K. What is the length of segment NE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center X and circumradius 9. Let W be the circumcircle of polygon ABCDE. Point V is the midpoint of segment BC. Let Z be the line through points C and V. Let Y be the reflection of circle W across line Z. Let H be the inversion of point E in circle Y. Find the measure of angle HXD, 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 ABCDEFHIJK be a regular 10-gon with center G and circumradius 5. Let W be the circumcircle of polygon ABCDEFHIJK. Point X is the midpoint of segment GI. Circle M is the reflection of circle W across point C. Let R be the inversion of point E in circle M. Let U be the midpoint of segment JX. What is the distance between points U and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QV = 11. Let ABCDEFG be a regular 7-gon with center V and circumradius 1. Let U be the midpoint of segment FA. Point S is the reflection of point G across point A. Point R is the reflection of point S across segment DA. Find the distance between points R and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center Q and circumradius 11. Line H is the angle bisector of angle EAD. Point R is the reflection of point C across line H. Line N is the angle bisector of angle REA. Line S is the reflection of line N across line H. Line M passes through point Q parallel to line S. Point P is the reflection of point D across line M. Let W be the reflection of point P across point D. Find the length of segment FW. 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 X and circumradius 12. Point G is the midpoint of segment CD. Let M be the line through point C perpendicular to segment EC. Point P is the reflection of point G across line M. Point Y is the reflection of point E across segment PG. What is the length of segment YG? 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 N and circumradius 1. Let T be the midpoint of segment JL. Let M be the midpoint of segment DN. Line U is the angle bisector of angle MHE. Point Q is the reflection of point T across line U. Point O is the reflection of point I across point H. What is the distance between points Q 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 ABCDEFHI be a regular octagon with center G and circumradius 1. Let T be the midpoint of segment BE. Line Z is the perpendicular bisector of segment GD. Point W is the reflection of point B across line Z. Let Q be the reflection of point T across point W. Point O is the midpoint of segment CB. Find 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]
Let ABCDEFGHI be a regular 9-gon with center V and circumradius 8. Let Q be the angle bisector of angle EVH. Let K be the reflection of point V across point B. Point T is the reflection of point C across point K. Let L be the reflection of line Q through point D. Point J is the reflection of point T across line L. What is the distance between points J 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 ABDE be a square with center C and circumradius 2. Let H be the line through points C and D. Let N be the circumcircle of polygon ABDE. Let F be the angle bisector of angle DEB. Line S is the reflection of line F across line H. Circle L is the reflection of circle N across line S. Point K is the inversion of point A with respect to circle L. What is 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 ABCDE be a regular 5-gon with center H and circumradius 12. Let L be the reflection of point C across point H. Let Y be the circumcircle of polygon ABCDE. Line R is the perpendicular bisector of segment CH. Let J be the reflection of line R through point C. Circle I is the reflection of circle Y across line J. Circle K is the reflection of circle I across line R. Let S be the inversion of point D in circle K. Point M is the midpoint of segment LS. Find the length of segment MB. 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 X and circumradius 5. Point F is obtained by rotating point X by angle pi about point C. Let R be the perpendicular bisector of segment BD. Let Q be the line through points F and A. Line N is the reflection of line Q across line R. Point U is the reflection of point X across line N. Find the length of segment UB. 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 4. Point V is the reflection of point E across point H. Line I is the perpendicular bisector of segment CB. Let Z be the line through point D parallel to line I. Point Y is the reflection of point V across line Z. What is the distance between points G 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 V and circumradius 3. Line Z passes through points V and G. Let S be the line through points H and A. Line X passes through point F parallel to line Z. Lines S and X intersect at point J. Let O be the midpoint of segment BH. Find the length of segment OJ. 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 H be the angle bisector of angle EBD. Let J be the line through point A perpendicular to line H. Point N is the reflection of point D across line J. Let O be the reflection of point D across point N. What is the length of segment CO? 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 1. Let X be the midpoint of segment HD. Let Z be the midpoint of segment XG. Let U be the angle bisector of angle NDZ. Line S is the reflection of line U across point Z. Point M is the midpoint of segment DX. Let R be the line through point M parallel to line S. Point V is the reflection of point G across line R. Find the distance between points E 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 ABCDF be a regular 5-gon with center E and circumradius 1. Line T passes through points A and D. Point O is obtained by rotating point C by angle 3pi/4 about point D. Let Y be the perpendicular bisector of segment OA. Line G passes through point B parallel to line Y. Point S is the intersection of lines T and G. Find the distance between points S and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center N and circumradius 10. Point J is the midpoint of segment CD. Point W is the reflection of point J across segment CB. Let Z be the line through points A and W. Point Q is the reflection of point C across line Z. Find the distance between points Q 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 ABCDEFGH be a regular octagon with center O and circumradius 11. Point R is the reflection of point G across point E. Let Q be the line through points C and R. Point W is the reflection of point A across line Q. Point S is the midpoint of segment WH. What is the length of segment SC? 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 H and circumradius 8. Point R is the midpoint of segment CH. Line G is the angle bisector of angle BCR. Point N is the reflection of point R across line G. Let Q be the midpoint of segment HR. Point J is the midpoint of segment RC. Let K be the midpoint of segment JA. Let L be the reflection of point K across point N. Find the distance between points Q and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCE is a square with center D and circumradius 2. Line N is the perpendicular bisector of segment ED. Let Q be the reflection of line N through point E. Point X is the reflection of point E across point A. Let O be the perpendicular bisector of segment BC. Let G be the reflection of line Q across line O. Segment BX and line G intersect at point S. What is the distance between points S 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 ABCDEFGHIJK be a regular 11-gon with center R and circumradius 3. Let W be the reflection of point K across point G. Line Y passes through points C and A. Let S be the reflection of point A across point W. Line L passes through point I perpendicular to line Y. Let Z be the circumcircle of polygon ABCDEFGHIJK. Circle V is the reflection of circle Z across line L. Let P be the inversion of point S in circle V. What is 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]
ABCDEF is a regular hexagon with center U and circumradius 9. Let N be the reflection of point U across point C. Let J be the reflection of point U across point A. Let W be the midpoint of segment NB. What is the distance between points W 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 S and circumradius 5. Let R be the angle bisector of angle AEF. Point X is the reflection of point C across line R. Let U be the circumcircle of polygon ABCDEF. Let O be the inversion of point X in circle U. Find the length of segment AO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center Y and circumradius 5. Point K is the midpoint of segment AY. Line O is the angle bisector of angle FGD. Let Q be the midpoint of segment BE. Let R be the reflection of line O through point D. Point T is the reflection of point K across line R. Let V be the midpoint of segment DQ. Point M is the reflection of point T across point V. Find the length of segment MY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center Q and circumradius 8. Let F be the perpendicular bisector of segment CB. Let L be the line through point B perpendicular to line F. Let O be the midpoint of segment BQ. Point Z is the reflection of point O across line L. Find the distance between points B and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center K and circumradius 6. Let U be the circumcircle of polygon ABCDEFGHIJ. Let S be the reflection of point A across point H. Circle N is the reflection of circle U across point I. Let T be the inversion of point S in circle N. 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 ABCDFGHI be a regular octagon with center E and circumradius 5. Let T be the midpoint of segment GF. Point Q is the midpoint of segment TE. Point P is the midpoint of segment TF. Find the measure of angle QPF, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center U and circumradius 7. Let P be the perpendicular bisector of segment EA. Let O be the line through points C and A. Line K is the reflection of line P across line O. Point S is the midpoint of segment AC. Point L is the reflection of point A across line K. Let R be the reflection of point S across point L. Let T be the reflection of point R across point E. Find the distance between points L and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center I and circumradius 9. Let R be the circumcircle of polygon ABCDEF. Line Z is the angle bisector of angle BFD. Point X is obtained by rotating point B by angle pi/4 about point E. Let Y be the reflection of circle R through point B. Point W is the inversion of point E with respect to circle Y. Circle V is the reflection of circle Y across point E. Point G is the inversion of point W with respect to circle V. Point J is the reflection of point G across line Z. Find 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]
ABCDEF is a regular hexagon with center M and circumradius 12. Let T be the midpoint of segment AC. Let Z be the midpoint of segment BT. Let P be the reflection of point Z across point F. What is the distance between points B 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 O and circumradius 9. Let U be the reflection of point B across point A. Line Z is the perpendicular bisector of segment UA. Point S is the reflection of point D across line Z. Point V is the midpoint of segment UO. Point Q is the reflection of point V across point C. What is the length of segment QS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHI is a regular 9-gon with center N and circumradius 2. Point Q is obtained by rotating point C by angle 7pi/12 about point E. Point X is the midpoint of segment BF. Let L be the midpoint of segment XG. Line O is the angle bisector of angle QNF. Point R is the reflection of point L across line O. What is the length of segment BR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFGHIJKL is a regular 11-gon with center C and circumradius 3. Let Z be the circumcircle of polygon ABDEFGHIJKL. Line N is the angle bisector of angle GBL. Line Y is the perpendicular bisector of segment HF. Let V be the reflection of line N across line Y. Let X be the reflection of circle Z across line N. Let U be the midpoint of segment GF. Let Q be the line through points I and J. Let O be the inversion of point U in circle X. Lines V and Q intersect at point R. Find the distance between points O and R. 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 T and circumradius 10. Line N passes through point F parallel to segment GI. Line U passes through point C parallel to line N. Point Q is the reflection of point B across line U. Find the length of segment QI. 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. Point R is the midpoint of segment AH. Let T be the circumcircle of polygon ABCDEGHIJK. Circle O is the reflection of circle T across point I. Point Q is the midpoint of segment CR. Point V is the inversion of point Q with respect to circle O. Point P is the midpoint of segment VK. What is the length of segment PG? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ACDEFG is a regular hexagon with center B and circumradius 10. Line L is the perpendicular bisector of segment DB. Point W is the reflection of point D across segment GA. Let O be the angle bisector of angle GBF. Let K be the line through point W parallel to line O. Line N is the reflection of line L across line K. Let M be the line through points A and B. Point S is the midpoint of segment CG. Line Q is the reflection of line M across point F. Lines Q and N intersect at point V. What is the length of segment VS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center S and circumradius 9. Let Y be the reflection of point F across point B. Let X be the perpendicular bisector of segment EY. Line J passes through point E parallel to line X. Point H is the reflection of point A across line J. What is the distance between points H and A? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 9. Let Z be the angle bisector of angle DMC. Point X is the midpoint of segment FM. Point Y is the reflection of point G across line Z. Point T is the reflection of point H across point X. Find the distance between points Y and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFGHIJKLM is a regular dodecagon with center A and circumradius 8. Point P is the reflection of point I across point B. Point N is obtained by rotating point L by angle 3pi/4 about point K. Line Z is the angle bisector of angle ACP. Let S be the line through point N parallel to line Z. Point Q is the reflection of point K across line S. Let T be the angle bisector of angle BQE. Point R is the reflection of point H across line T. Find the distance between points E and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFGHIJKLM is a regular dodecagon with center C and circumradius 6. Point N is obtained by rotating point F by angle 3pi/4 about point K. Let X be the circumcircle of polygon ABDEFGHIJKLM. Let V be the angle bisector of angle NDI. Let T be the reflection of circle X across line V. Let Y be the inversion of point L in circle T. What is the distance between points D and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center V and circumradius 9. Let W be the perpendicular bisector of segment VA. Let N be the reflection of line W through point A. Point R is the reflection of point E across line N. What is the distance between points A 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 ABCDE be a regular 5-gon with center N and circumradius 9. Line T passes through points A and B. Let G be the reflection of line T through point E. Line Z passes through point A perpendicular to line G. Point L is the reflection of point D across line Z. What is the length of segment AL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center J and circumradius 2. Let K be the circumcircle of polygon ABCDEF. Let T be the reflection of point E across point C. Let Z be the line through points D and T. Let Q be the reflection of circle K through point C. Let P be the reflection of circle Q across line Z. Point O is the inversion of point C with respect to circle P. What is the distance between points T 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 ABCDEFHIJKLM be a regular dodecagon with center G and circumradius 1. Let W be the midpoint of segment IA. Point Q is the midpoint of segment LM. Let N be the midpoint of segment EQ. Find the length of segment NW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 12. Point X is the reflection of point D across point H. Line W passes through points J and X. Let S be the perpendicular bisector of segment GH. Let R be the reflection of line W across line S. Point Z is the reflection of point U across line R. 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]
ABCDEF is a regular hexagon with center K and circumradius 3. Let H be the perpendicular bisector of segment KA. Point Q is the reflection of point D across line H. Let U be the reflection of point A across point E. Let R be the midpoint of segment EU. Find the distance between points R 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 BCDEFGHIJKLM be a regular dodecagon with center A and circumradius 9. Point Y is obtained by rotating point B by angle pi/6 about point J. Line O is the angle bisector of angle JIK. Point U is the reflection of point H across line O. Let N be the angle bisector of angle GAU. Point V is the reflection of point D across line N. Find the distance between points V and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center W and circumradius 4. Let Y be the angle bisector of angle WCD. Point S is the reflection of point F across line Y. Line T passes through points D and F. Point N is the reflection of point S across segment BC. Point P is the reflection of point B across line T. What is the length of segment PN? 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 Q and circumradius 11. Line Z is the perpendicular bisector of segment DA. Let M be the line through point A parallel to line Z. Let O be the line through point D parallel to line M. Point R is the reflection of point B across line O. Point X is the midpoint of segment ER. 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]
ABCDEFGHIJK is a regular 11-gon with center Q and circumradius 8. Let N be the midpoint of segment HD. Let Y be the perpendicular bisector of segment NK. Point Z is the reflection of point A across line Y. Point O is the reflection of point Z across point D. What is the length of segment OZ? 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 V and circumradius 9. Point P is the reflection of point A across point E. Point N is the reflection of point C across point E. Line U passes through points A and N. Point S is the reflection of point P across line U. Point K is the midpoint of segment BD. Find the length of segment SK. 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 J and circumradius 4. Let Q be the circumcircle of polygon ABCD. Line V is the angle bisector of angle DCJ. Point U is obtained by rotating point D by angle 3pi/4 about point J. Let N be the rotation of point C by angle 11pi/12 around point A. Point G is the reflection of point U across line V. Point O is the inversion of point G with respect to circle Q. Let R be the reflection of circle Q across line V. Let X be the inversion of point O in circle R. What is the measure of angle BNX, 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 W and circumradius 8. Let O be the perpendicular bisector of segment BW. Let V be the line through point E perpendicular to line O. Let L be the reflection of point F across point E. Let G be the reflection of point F across point A. Line X is the angle bisector of angle LFG. Let I be the reflection of line V across line X. Point Y is the reflection of point G across line I. 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]
Let ABCDF be a regular 5-gon with center E and circumradius 4. Line P passes through points F and E. Line S is the reflection of line P across point B. Line X passes through points B and C. Point Q is the intersection of lines X and S. Find the distance between points A 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 ABCDEFGHIJKL be a regular dodecagon with center O and circumradius 9. Line M is the angle bisector of angle DBG. Point V is the reflection of point A across line M. Point W is the midpoint of segment CH. What is the length of segment VW? 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 4. Let U be the rotation of point N by angle pi/2 around point D. Line L is the perpendicular bisector of segment UN. Line J passes through point E perpendicular to line L. Let V be the perpendicular bisector of segment CD. Line P is the reflection of line J across line V. Point T is the reflection of point N across line P. Find the measure of angle ABT, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center M and circumradius 4. Let X be the line through point M parallel to segment EB. Point U is the midpoint of segment EB. Point I is the midpoint of segment UE. Point G is the reflection of point I across line X. What is the length of segment GB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEGHI be a regular octagon with center F and circumradius 4. Let X be the reflection of point E across point B. Let Q be the circumcircle of polygon ABCDEGHI. Let V be the angle bisector of angle CFX. Line R passes through point X perpendicular to line V. Let P be the midpoint of segment FG. Point U is the reflection of point H across line R. Point J is the inversion of point P with respect to circle Q. Find the distance between points J and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center Y and circumradius 2. Point X is the reflection of point B across segment FA. Let P be the angle bisector of angle YCI. Line T is the reflection of line P across point F. Let R be the reflection of point D across point H. Line Q passes through point K parallel to line T. Point U is the reflection of point J across line Q. What is the measure of angle XUR, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center R and circumradius 11. Let I be the angle bisector of angle GEA. Point Q is the reflection of point G across line I. Let W be the line through point E perpendicular to line I. Point K is the reflection of point Q across line W. What is the length of segment KD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFG is a regular hexagon with center D and circumradius 12. Line U passes through points G and A. Let L be the line through point C parallel to segment EA. Point M is the intersection of lines L and U. Point Q is the reflection of point D across point F. Find the distance between points Q 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 K and circumradius 2. Let R be the line through points B and K. Line N is the reflection of line R across point D. Let T be the circumcircle of polygon ABCDEF. Let S be the reflection of line N through point K. Let J be the reflection of circle T across line S. Let O be the inversion of point B in circle J. Point U is the inversion of point O with respect to circle T. What is the length of segment UF? 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 5. Point S is obtained by rotating point G by angle 5pi/12 about point D. Point T is the intersection of segments FS and JC. Let U be the midpoint of segment TB. Let W be the circumcircle of polygon ABCDEFGHIJ. Let V be the inversion of point U in circle W. What is the length of segment GV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJLM be a regular dodecagon with center K and circumradius 11. Let Y be the circumcircle of polygon ABCDEFGHIJLM. Point V is the reflection of point B across segment MG. Let O be the midpoint of segment KA. Point P is the inversion of point V with respect to circle Y. What is the distance between points P 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 ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 9. Let Q be the line through points J and F. Point Y is the reflection of point I across line Q. Let W be the midpoint of segment LY. Let R be the midpoint of segment MW. What is the length of segment ER? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFGHIJKL is a regular 11-gon with center D and circumradius 11. Let S be the circumcircle of polygon ABCEFGHIJKL. Let U be the reflection of circle S through point L. Let W be the perpendicular bisector of segment EF. Circle O is the reflection of circle U across line W. Point Z is the inversion of point I with respect to circle O. Find the distance between points C and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center M and circumradius 11. Let J be the reflection of point F across point D. Point V is the midpoint of segment CE. Let P be the reflection of point J across point V. What is the distance between points F 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 L and circumradius 9. Line T is the perpendicular bisector of segment DL. Let N be the perpendicular bisector of segment CB. Let S be the reflection of line N across line T. Point X is the reflection of point F across line S. What is the measure of angle XDL, in radians? 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 4. Let T be the reflection of point F across point K. Point N is the midpoint of segment TM. Point Y is the reflection of point N across segment BL. Find the distance between points Y and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CK = 5. ABDEFGHIJLMN is a regular dodecagon with center K and circumradius 10. Let X be the line through points J and I. Line S is the reflection of line X across point M. Let Q be the reflection of line S through point K. Point V is the intersection of segment JG and line Q. What is the length of segment VE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGIJ is a regular 9-gon with center H and circumradius 9. Line S is the angle bisector of angle DJC. Point X is the reflection of point D across line S. Let V be the midpoint of segment GB. Point Z is the midpoint of segment XV. Find the length of segment ZJ. 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 10. Point T is the reflection of point D across point B. Line Y passes through point T parallel to segment AD. Point R is the reflection of point E across line Y. What is the length of segment CR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center N and circumradius 3. Line Q passes through points E and H. Line K passes through point G parallel to segment AH. Point S is the intersection of lines K and Q. 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]
ABCDEFGHIJKL is a regular dodecagon with center T and circumradius 9. Let Z be the midpoint of segment AC. Segments HC and BF intersect at point Y. Find the length of segment ZY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center S and circumradius 11. Let M be the midpoint of segment EF. Point L is the reflection of point M across point C. Point R is the reflection of point B across segment EF. Find the length of segment LR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEG is a regular hexagon with center F and circumradius 3. Let M be the circumcircle of polygon ABCDEG. Line R is the angle bisector of angle DCG. Point K is on line R. Let J be the inversion of point K in circle M. What is the distance between points D and J? 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 Q and circumradius 1. Let S be the circumcircle of polygon ABCDEFGH. Line W passes through points D and B. Circle O is the reflection of circle S across line W. Let L be the inversion of point F in circle O. What is the length of segment LB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center X and circumradius 12. Let Z be the line through point C parallel to segment AK. Let P be the midpoint of segment AK. Let O be the midpoint of segment LD. Point V is the reflection of point P across line Z. Point Y is the reflection of point V across point O. What is the distance between points Y and F? 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 V and circumradius 10. Let W be the line through points B and C. Point Q is the reflection of point G across line W. Point X is the reflection of point F across segment BD. What is the distance between points Q and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center M and circumradius 9. Let P be the circumcircle of polygon ABCDEF. Let V be the midpoint of segment ED. Let S be the reflection of circle P through point B. Line I is the perpendicular bisector of segment BV. Circle X is the reflection of circle S across point V. Line R passes through point M parallel to line I. Segment BV and line R intersect at point T. Point J is the inversion of point T with respect to circle X. What is the distance between points J and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center T and circumradius 10. Line U is the perpendicular bisector of segment GF. Let Q be the reflection of point D across point G. Let S be the midpoint of segment FC. Point L is the reflection of point S across line U. What is the distance between points Q and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BCDEFGH be a regular 7-gon with center A and circumradius 7. Let I be the line through points D and E. Line O passes through point G parallel to line I. Point Z is the reflection of point C across line O. Let P be the rotation of point C by angle 2pi/3 around point G. Line N is the reflection of line I across point Z. Let Y be the angle bisector of angle FPD. Let T be the reflection of line N across line Y. Point R is the intersection of lines I and T. Find the length of segment FR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEGH is a regular 7-gon with center F and circumradius 6. Line Y passes through points D and G. Let R be the line through points F and C. Point O is the intersection of lines R and Y. Let P be the reflection of point E across point A. Find the distance between points P and O. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEG is a regular hexagon with center F and circumradius 5. Let W be the line through points E and C. Point N is the reflection of point E across point C. Line R passes through point B perpendicular to line W. Let H be the line through point F perpendicular to line R. Point U is the reflection of point N across line H. Find the length of segment UB. 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 3. Line X is the perpendicular bisector of segment DC. Let Q be the line through point D parallel to line X. Let P be the reflection of line Q across line X. Let T be the reflection of line X across line P. Point K is the reflection of point A across line T. What is the distance between points K and E? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center U and circumradius 6. Let Y be the rotation of point E by angle 5pi/6 around point F. Point T is the midpoint of segment YU. Let J be the midpoint of segment BT. Point M is the midpoint of segment JT. Point O is the midpoint of segment FE. Find 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 N and circumradius 8. Point K is the reflection of point E across point A. Point M is the reflection of point E across point D. Line V passes through points F and D. Point T is the reflection of point K across line V. What is the length of segment TM? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 8. Let S be the perpendicular bisector of segment IH. Let Y be the reflection of line S through point D. Line R is the reflection of line Y across point F. Point T is the reflection of point L across point G. Point X is the reflection of point G across line R. What is the distance between points T and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHI be a regular 9-gon with center Z and circumradius 9. Let U be the midpoint of segment ZG. Let Q be the angle bisector of angle ACI. Let O be the line through point A parallel to line Q. Let M be the reflection of line O through point U. Point N is the reflection of point Z across line M. Find the distance between points F and N. 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 L and circumradius 5. Point G is the midpoint of segment AD. Let T be the reflection of point C across point E. Let Y be the reflection of point G across point T. Find the length of segment YC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center Q and circumradius 9. Let P be the line through points D and H. Line Y passes through point G perpendicular to line P. Point Z lies on line Y. Line I passes through points D and A. Point U is the reflection of point Z across line I. What is the measure of angle FAU, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEF is a regular 5-gon with center C and circumradius 4. Let J be the reflection of point F across point A. Line M is the perpendicular bisector of segment BF. Line Y passes through point J parallel to line M. Let G be the line through point D perpendicular to line Y. Point N is the reflection of point F across line G. Find the length of segment JN. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFHIJ be a regular 9-gon with center G and circumradius 7. Let K be the circumcircle of polygon ABCDEFHIJ. Point N is the midpoint of segment CI. Point R is the midpoint of segment EH. Let Z be the line through point F perpendicular to segment CI. Let S be the reflection of circle K across line Z. Point X is the inversion of point I with respect to circle S. Point U is the midpoint of segment NX. What is the distance between points R and U? 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 O and circumradius 2. Line N is the angle bisector of angle CGB. Let L be the line through point C parallel to line N. Let J be the line through point H parallel to line L. Point M is the reflection of point A across line J. 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]
Let ABCDEFG be a regular 7-gon with center M and circumradius 5. Let K be the line through points M and G. Let S be the line through point A parallel to line K. Point T is the reflection of point F across point E. Point P is the reflection of point T across line S. Point H is the reflection of point P across segment TG. Find the distance between points C 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 ABCDEF be a regular hexagon with center Y and circumradius 3. Point Q is obtained by rotating point F by angle pi/6 about point C. Line Z is the angle bisector of angle BAF. Line I is the reflection of line Z across point B. Point U is the reflection of point A across point F. Point V is the reflection of point Q across line I. What is the distance between points V and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJ be a regular 10-gon with center V and circumradius 3. Let W be the rotation of point J by angle 7pi/12 around point I. Let M be the midpoint of segment DW. Point Y is the reflection of point W across point J. Line O passes through point G parallel to segment JV. Line N passes through point M perpendicular to segment JV. Point P is the reflection of point C across line N. Point X is the reflection of point P across segment JV. Let R be the reflection of line N through point Y. Point T is the intersection of lines R and O. Find the distance between points X and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center S and circumradius 12. Point Z is the midpoint of segment DA. Point G is the reflection of point B across segment CE. Let P be the midpoint of segment CG. Point R is the reflection of point A across point P. What is the length of segment RZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center N and circumradius 4. Line H passes through points D and B. Let J be the line through points A and B. Line R passes through point F parallel to line H. Lines J and R intersect at point Z. Let V be the circumcircle of polygon ABCDEF. Let I be the inversion of point Z in circle V. What is the distance between points I and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFG is a regular hexagon with center A and circumradius 3. Let V be the reflection of point A across point E. Line X is the angle bisector of angle VGD. Let I be the reflection of line X through point D. Point M is the reflection of point E across line I. What is the distance between points G 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 ABCDEFGH be a regular octagon with center I and circumradius 3. Point U is on segment FB. Point Z is the reflection of point A across segment DC. What is the measure of angle UBZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center N and circumradius 4. Line M passes through point D perpendicular to segment CB. Point U is the reflection of point N across point D. Line E is the reflection of line M across point N. Point J is the reflection of point N across line E. What is the distance between points J and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDE be a square with center C and circumradius 7. Line W is the perpendicular bisector of segment AC. Point R is the midpoint of segment DB. Point S is the reflection of point R across line W. Let I be the reflection of point E across point B. Find the distance between points S 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 Y and circumradius 4. Point W is the midpoint of segment DG. Let L be the rotation of point C by angle 5pi/6 around point B. Line R is the perpendicular bisector of segment BW. Let Z be the midpoint of segment LC. Point V is the reflection of point E across line R. What is the distance between points Z and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJK is a regular 11-gon with center P and circumradius 10. Let X be the circumcircle of polygon ABCDEFGHIJK. Point N is the reflection of point C across point F. Let V be the inversion of point N in circle X. Point O is the midpoint of segment VP. What is the length of segment IO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center J and circumradius 8. Let P be the reflection of point B across point D. Point Q is the midpoint of segment PE. Point N is the reflection of point B across segment QC. What is the measure of angle NDE, in radians? 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 I and circumradius 4. Let O be the circumcircle of polygon ABCDE. Let U be the reflection of circle O through point D. Point T is the midpoint of segment BC. Let J be the inversion of point I in circle U. What is the length of segment TJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center Z and circumradius 7. Point U is the midpoint of segment BZ. Let W be the reflection of point U across point C. Point K is the midpoint of segment WZ. Find the length of segment KA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center T and circumradius 6. Let K be the rotation of point F by angle 3pi/4 around point G. Let Z be the angle bisector of angle DKB. Line Y is the reflection of line Z across point J. Line V is the reflection of line Y across point H. Point Q is the reflection of point F across line V. What is the length of segment DQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center S and circumradius 8. Let F be the angle bisector of angle SBD. Line P passes through points C and E. Line Z is the reflection of line P across line F. Lines P and Z intersect at point J. What is the distance between points J and E? 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 4. Point O is the midpoint of segment AB. Point P is the reflection of point H across point F. Line J is the perpendicular bisector of segment FO. Point V is the reflection of point C across line J. What is the distance between points V and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center H and circumradius 5. Point J is obtained by rotating point B by angle pi/12 about point D. Point F is obtained by rotating point D by angle 3pi/4 about point J. Let R be the circumcircle of polygon ABCD. Line M is the perpendicular bisector of segment BD. Point S is the reflection of point F across line M. Point V is the inversion of point S with respect to circle R. Point G is the reflection of point J across point S. Find the distance between points V 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 ABCDEFGHIKL be a regular 11-gon with center J and circumradius 6. Let X be the line through points A and C. Let M be the line through point I parallel to line X. Segment LH and line M intersect at point Q. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center O and circumradius 3. Point Y is obtained by rotating point C by angle pi/12 about point D. Let L be the reflection of point B across point A. Line R passes through point H parallel to segment FE. Point N is on segment FE. Line P is the reflection of line R across point N. Point X is the reflection of point L across line P. Point U is the reflection of point Y across point X. Find the length of segment UF. 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 11. Line O passes through points L and H. Point Q is the intersection of line O and segment KE. Point X is the midpoint of segment GJ. What is the distance between points X 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 M and circumradius 2. Point P is the midpoint of segment EA. Point Z is the reflection of point B across point E. Let H be the midpoint of segment ZP. Point Q is the midpoint of segment AH. What is 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]
ABCDEFGHIJ is a regular 10-gon with center Z and circumradius 11. Let M be the midpoint of segment IF. Point N is the midpoint of segment EM. Point X is the reflection of point F across point J. What is the length of segment XN? 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 N and circumradius 6. Point X is the reflection of point B across point C. Let O be the angle bisector of angle DNE. Let Z be the line through point X perpendicular to line O. Let U be the line through point N parallel to line Z. Point H is the reflection of point C across line U. Let J be the reflection of point H across point A. What is the distance between points C and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHJK is a regular 10-gon with center I and circumradius 5. Let P be the circumcircle of polygon ABCDEFGHJK. Let W be the reflection of circle P through point G. Let V be the inversion of point A in circle W. Let R be the angle bisector of angle FVE. Point N is the reflection of point J across line R. What is the length of segment HN? 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 4. Let K be the line through points B and F. Let N be the perpendicular bisector of segment FA. Line U is the reflection of line N across point E. Line J is the reflection of line K across point E. Line V passes through points D and C. Point R is the reflection of point C across line U. Point W is the intersection of lines V and J. What is the distance between points R and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center V and circumradius 12. Z is placed 10 units away from E. Let S be the perpendicular bisector of segment BV. Let X be the midpoint of segment BV. Let P be the reflection of point Z across point X. Point G is the reflection of point P across segment DE. Point W is the reflection of point G across line S. Find the distance between points W 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 ABCDE be a regular 5-gon with center U and circumradius 10. Line N passes through points A and B. Let Z be the reflection of line N through point C. Let H be the reflection of line Z through point U. Line Y passes through point C perpendicular to line H. Point X is the reflection of point A across line Y. What is the distance between points X and E? 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 9. Line O is the angle bisector of angle JGB. Line V is the reflection of line O across point E. Point M is the reflection of point F across line V. Let S be the midpoint of segment HM. Point Q is the reflection of point G across point S. What is the distance between points Q and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFGHIJK is a regular 10-gon with center D and circumradius 11. Line M passes through points A and I. Let L be the line through point B perpendicular to line M. Line N passes through point G parallel to line L. Point X is the reflection of point H across line N. Find the measure of angle HGX, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center Q and circumradius 2. Let Y be the line through points C and Q. Let N be the reflection of point H across point G. Let Z be the line through points N and E. Lines Z and Y intersect at point P. Find the distance between points P 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 ABCDEFGH be a regular octagon with center X and circumradius 12. Let U be the midpoint of segment GH. Let Y be the midpoint of segment HU. Line T is the perpendicular bisector of segment YE. Point I is the reflection of point U across line T. Find the length of segment CI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center K and circumradius 2. Let I be the perpendicular bisector of segment AB. Line Y passes through point K perpendicular to line I. Line R is the reflection of line Y across point B. Line L is the reflection of line Y across line R. Let J be the line through points E and K. Point H is the reflection of point B across line L. Point G is the reflection of point H across line J. What is the length of segment KG? 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 Q be the reflection of point Y across point E. Let O be the rotation of point C by angle pi/4 around point B. Point W is the midpoint of segment FD. Let S be the line through point Q parallel to segment OW. Point K is on line S. Let R be the reflection of point K across point B. Let I be the reflection of point R across point F. Line J passes through point K perpendicular to segment OW. Point V is the reflection of point I across line J. What is the distance between points V and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center Q and circumradius 3. Let P be the midpoint of segment DA. Let Z be the circumcircle of polygon ABCDE. Let Y be the inversion of point P in circle Z. Point L is the midpoint of segment BY. What is the distance between points P 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 L and circumradius 7. Let K be the midpoint of segment CB. Point J is the reflection of point F across segment CB. Let P be the midpoint of segment LJ. Point U is the reflection of point K across point F. Find the distance between points U and P. 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 V and circumradius 5. Let O be the perpendicular bisector of segment CH. Let U be the midpoint of segment HE. Line K passes through point D parallel to line O. Point Z is the reflection of point U across line K. Find the length of segment ZG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center X and circumradius 3. Let J be the perpendicular bisector of segment HG. Let T be the reflection of point F across point G. Let K be the line through point H perpendicular to line J. Point S is the reflection of point X across line K. Find the measure of angle XST, 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 6. Line P passes through points D and A. Line Q passes through points C and E. Line L is the reflection of line P across line Q. Point Y is the reflection of point D across line L. What is the distance between points G and Y? 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. Let M be the midpoint of segment AB. Let Z be the reflection of point E across point A. Point O is the reflection of point E across segment ZM. What is the distance between points O and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFGHIJK is a regular 10-gon with center D and circumradius 8. Line M is the angle bisector of angle GHD. Line Z passes through point G parallel to line M. Let S be the circumcircle of polygon ABCEFGHIJK. Let Y be the reflection of line Z across line M. Point V is the reflection of point G across line Y. Let Q be the inversion of point V in circle S. What is the distance between points C and Q? 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 F and circumradius 11. Point E is the reflection of point F across point A. Point G is the reflection of point E across point F. Point V is the reflection of point G across segment EB. What is the distance between points C and V? 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 V and circumradius 1. Point S is the midpoint of segment EG. Let T be the circumcircle of polygon ABCDEFGHIJ. Let R be the reflection of circle T through point F. Let Q be the inversion of point G in circle R. What is the distance between points Q and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center L and circumradius 5. Point S is the midpoint of segment BD. Line K passes through points A and S. Let O be the midpoint of segment CB. Let M be the reflection of line K through point O. Point X is the reflection of point B across line M. Find the length of segment XB. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFHIJKLM be a regular dodecagon with center G and circumradius 2. Let W be the circumcircle of polygon ABCDEFHIJKLM. Line N is the angle bisector of angle EFG. Point Q is the reflection of point E across line N. Point S is the inversion of point Q with respect to circle W. Find the distance between points E and S. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFG be a regular hexagon with center B and circumradius 2. Let S be the circumcircle of polygon ACDEFG. Let O be the midpoint of segment GE. Point Y is the midpoint of segment BD. Let P be the inversion of point Y in circle S. Find the length of segment PO. 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 9. Point R is the reflection of point E across point D. Let I be the circumcircle of polygon ABCDE. Line X is the perpendicular bisector of segment RC. Circle Y is the reflection of circle I across line X. Point O is the inversion of point C with respect to 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]
Let ABCDE be a regular 5-gon with center G and circumradius 1. Let L be the line through points E and D. Let R be the reflection of line L through point G. Point K is the reflection of point C across point G. Point Z is the reflection of point C across line R. What is the length of segment ZK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDF is a regular 5-gon with center E and circumradius 7. Line P is the perpendicular bisector of segment BA. Point R is the reflection of point F across segment BA. Point U is the reflection of point R across line P. What is the measure of angle RUD, in radians? 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 4. Let R be the line through points D and C. Let O be the reflection of line R through point A. Let U be the line through point J perpendicular to line O. Point T is the reflection of point G across line U. Find the distance between points T and E. 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 S and circumradius 11. Line R is the perpendicular bisector of segment AS. Let V be the circumcircle of polygon ABCDEFG. Circle L is the reflection of circle V across line R. Let H be the reflection of circle L through point B. Point Z is the inversion of point A with respect to circle H. Find the length of segment ZD. 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 Q and circumradius 1. Let X be the midpoint of segment QG. Line O is the angle bisector of angle CFX. Let M be the reflection of line O through point X. Point U is the reflection of point Q across line M. Find the length of segment AU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center Y and circumradius 3. Point W is obtained by rotating point F by angle pi/4 about point B. Let L be the midpoint of segment CB. Let T be the line through points G and L. Let Q be the rotation of point B by angle pi/4 around point F. Point N is the midpoint of segment YQ. Line Z is the perpendicular bisector of segment NE. Let S be the line through point F parallel to line Z. Lines T and S intersect at point U. Point P is the reflection of point W across line Z. Point X is the reflection of point A across segment UH. What is the distance between points X 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 W and circumradius 5. Point T is the reflection of point W across segment BA. Point M is the midpoint of segment BA. Line Z passes through points D and M. Point P is the reflection of point B across line Z. What is the distance between points P and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFG be a regular hexagon with center B and circumradius 5. Line W passes through points B and E. Let Z be the angle bisector of angle DEA. Point N is the midpoint of segment BD. Let J be the reflection of line W through point D. Line L is the reflection of line J across line Z. Point U is the reflection of point N across line L. What is the distance between points U and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFHIJKL be a regular 11-gon with center G and circumradius 5. Let U be the reflection of point J across point G. Let V be the angle bisector of angle UJH. Let N be the reflection of line V through point U. Point R is the reflection of point F across line N. 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]
Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 7. Let S be the rotation of point J by angle pi around point A. Let X be the midpoint of segment KA. Let Q be the reflection of point S across point I. Point Z is the midpoint of segment FX. Find the length of segment ZQ. 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 W and circumradius 4. Let Q be the rotation of point I by angle 5pi/12 around point K. Let S be the line through points J and A. Let N be the reflection of point W across point A. Let T be the line through point N parallel to line S. Point U is the reflection of point Q across line T. Let O be the line through point W parallel to segment CH. Point Z is the reflection of point C across point B. Point V is the reflection of point U across line O. What is the length of segment VZ? 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 Q and circumradius 2. Point Y is the reflection of point F across point G. Point T is the midpoint of segment YE. Let U be the midpoint of segment GT. Point M is the reflection of point B across segment TU. Find the length of segment MB. 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 11. Line W is the angle bisector of angle AHB. Let J be the line through point G parallel to line W. Segment AE and line J intersect at point Z. Point N is the reflection of point D across line W. Let Y be the reflection of point N across point Z. What is the length of segment YE? 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 2. Let L be the angle bisector of angle CFE. Line T is the reflection of line L across point C. Let Z be the reflection of line L across line T. Let S be the reflection of point B across point C. Point N is the reflection of point E across line Z. What is the length of segment NS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center Q and circumradius 6. Let H be the perpendicular bisector of segment DC. Let V be the reflection of line H through point C. Point O is the reflection of point E across line V. Point J is the reflection of point O across point D. What is the length of segment JD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDE be a square with center C and circumradius 6. Let L be the reflection of point D across point B. Point M is the reflection of point C across point E. Point T is the midpoint of segment LM. What is the distance between points L and T? 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 12. Let R be the rotation of point C by angle 5pi/12 around point B. Let O be the line through points D and F. Point J is the reflection of point C across line O. Line P is the angle bisector of angle JCE. Point V is the reflection of point R across line P. What is the distance between points E and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center G and circumradius 9. Let I be the line through points C and D. Point R is the midpoint of segment FA. Point T is the reflection of point G across line I. Let Q be the midpoint of segment TR. What is the length of segment CQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFGHI be a regular octagon with center C and circumradius 6. Point X is the midpoint of segment GD. Line Y is the perpendicular bisector of segment IC. Line J passes through point D parallel to line Y. Line M passes through points D and B. Line T passes through point C parallel to line M. Lines T and J intersect at point L. What is the measure of angle XLE, 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 ABCDEFGH be a regular octagon with center N and circumradius 5. Point Z is obtained by rotating point A by angle pi/6 about point F. Point V is the reflection of point N across segment FZ. Point T is the intersection of segments VG and FZ. Find the length of segment TC. Express your answer as a decimal number rounded to 2 decimal places.

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