[User]
Let ACDEFGHI be a regular octagon with center B and circumradius 12. Line K is the perpendicular bisector of segment EC. Point U is the midpoint of segment HB. Line V is the reflection of line K across point I. Point Z is on line V. Let J be the midpoint of segment UZ. Find the measure of angle DCJ, 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 ABCDEFGHIJKL be a regular dodecagon with center W and circumradius 1. Point Y is obtained by rotating point D by angle 2pi/3 about point E. Point N is the reflection of point G across point C. Let O be the line through points H and A. Let T be the perpendicular bisector of segment CN. Point X is the reflection of point J across line O. Point M is the reflection of point Y across line T. Find the distance between points X 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 Y and circumradius 6. Point Z is obtained by rotating point E by angle 3pi/4 about point F. Let J be the circumcircle of polygon ABCDEFGH. Circle Q is the reflection of circle J across point C. Let O be the line through points E and Z. Let S be the reflection of circle Q across line O. Point W is the inversion of point C with respect to circle S. Find the distance between points Y and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center Y and circumradius 12. Let J be the line through points D and H. Let X be the reflection of point C across point H. Point M is the midpoint of segment CF. Point Z is the reflection of point M across line J. Find the length of segment XZ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center Y and circumradius 1. Point O is obtained by rotating point C by angle 11pi/12 about point D. Point G is the midpoint of segment DC. Let U be the midpoint of segment OG. Point X is the midpoint of segment ED. Point H is the reflection of point Y across point X. Find the length of segment UH. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center L and circumradius 12. Point Q is the midpoint of segment CE. Point M is the reflection of point Q across point F. Point N is the reflection of point B across segment DC. Find the length of segment MN. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 12. Point T is obtained by rotating point D by angle pi/4 about point J. Let P be the midpoint of segment KG. Point Q is the midpoint of segment TL. Let X be the circumcircle of polygon ABCDEFGHIJKM. Point W is the inversion of point P with respect to circle X. What is the length of segment WQ? 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 2. Point Y is the midpoint of segment KC. Line M is the perpendicular bisector of segment YG. Point O is the reflection of point F across line M. Let R be the midpoint of segment YG. Point V is the reflection of point O across segment YG. What is the length of segment RV? 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 7. Point I is obtained by rotating point E by angle 2pi/3 about point G. Line N is the angle bisector of angle GBI. Let X be the reflection of line N through point H. Point Z is the reflection of point A across line X. Let M be the reflection of point Z across point Y. What is the length of segment DM? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center W and circumradius 7. Let Q be the perpendicular bisector of segment HJ. Let V be the line through point W perpendicular to line Q. Let U be the reflection of line V through point E. Point P is the reflection of point A across line U. What is 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 ABCDEF be a regular hexagon with center L and circumradius 11. Let I be the perpendicular bisector of segment EC. Let U be the midpoint of segment FD. Point X is the reflection of point U across line I. Find the length of segment AX. 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. Line N is the angle bisector of angle CHG. Line U passes through point B perpendicular to line N. Let W be the line through point D parallel to line U. Point Z is the reflection of point E across line W. Find the length of segment GZ. 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 U and circumradius 3. Point O is the reflection of point A across point C. Let Y be the midpoint of segment BO. Point N is the midpoint of segment CY. Point T is the midpoint of segment CN. What is the length of segment TD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center M and circumradius 12. Line Z passes through points D and E. Let I be the midpoint of segment DA. Point Q is the reflection of point B across line Z. Let N be the midpoint of segment AQ. What is the length of segment IN? 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 1. Let P be the perpendicular bisector of segment DB. Point L is the midpoint of segment DA. Point T is the reflection of point L across line P. Point Y is the reflection of point A across point B. Find the length of segment TY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEGHI is a regular octagon with center F and circumradius 9. Let Y be the angle bisector of angle BID. Point Q is the reflection of point H across line Y. Line U is the reflection of line Y across point F. Let P be the reflection of point Q across point H. Line U and segment AG intersect at point T. Point R is the reflection of point T across point E. What is the length of segment RP? 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 U and circumradius 8. Let Q be the midpoint of segment UE. Let X be the midpoint of segment HB. Let O be the circumcircle of polygon ABCDEFGH. Point L is obtained by rotating point Q by angle 11pi/12 about point B. Line M passes through points B and L. Line Y is the angle bisector of angle DEX. Circle P is the reflection of circle O across line Y. Line M and segment CG intersect at point J. Point V is the inversion of point C with respect to circle P. Find the length of segment JV. 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 11. Let R be the reflection of point M across point D. Line O is the angle bisector of angle IDB. Line Y passes through points R and K. Let X be the reflection of line O through point G. Line Q passes through point L perpendicular to line X. Line T passes through point B perpendicular to line X. Let S be the reflection of line O across line Y. Let P be the reflection of line Q across line T. Point V is the intersection of lines P and S. What is the distance between points A 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 ABCDEFGHIJ be a regular 10-gon with center X and circumradius 11. Point K is the midpoint of segment XH. Let V be the circumcircle of polygon ABCDEFGHIJ. Let O be the angle bisector of angle XAH. Point L is the reflection of point E across line O. Let W be the reflection of circle V through point K. Let Z be the inversion of point L in circle W. Find the distance between points F and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center Z and circumradius 11. Point Y is the reflection of point A across point D. Point W is the reflection of point E across point Y. Point K is the midpoint of segment DB. Let H be the reflection of point K across point Z. Find the length of segment WH. 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 6. Point K is the midpoint of segment GC. Point N is obtained by rotating point G by angle 3pi/4 about point F. Line Y is the perpendicular bisector of segment GD. Let X be the reflection of line Y through point A. Line Z passes through points F and N. Lines X and Z intersect at point U. Find the length of segment KU. 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 3. Let G be the circumcircle of polygon ABCDF. Line T is the angle bisector of angle DFE. Point H is the reflection of point F across point C. Let P be the reflection of circle G across line T. Let Y be the inversion of point C in circle P. Find the length of segment HY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFG be a regular 7-gon with center U and circumradius 9. Point Y is obtained by rotating point C by angle pi/4 about point A. Let N be the midpoint of segment BG. Line X passes through points D and C. Let I be the line through point Y parallel to line X. Point Q is the reflection of point N across line I. What is 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]
Let ABCDEFGHIJ be a regular 10-gon with center L and circumradius 9. Let R be the reflection of point L across point B. Line V passes through points D and E. Let S be the circumcircle of polygon ABCDEFGHIJ. Let P be the reflection of circle S across line V. Point U is the inversion of point R with respect to circle P. Point O is the inversion of point U with respect to circle S. What is the length of segment EO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center P and circumradius 10. Let Z be the circumcircle of polygon ABCDEFGH. Let S be the perpendicular bisector of segment CD. Let I be the reflection of circle Z through point G. Line N passes through point D parallel to line S. Line X is the reflection of line N across point H. Let Y be the inversion of point F in circle I. Point K is the reflection of point C across line X. What is the length of segment YK? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEF be a regular 5-gon with center B and circumradius 4. Let W be the line through points E and C. Let V be the line through point B parallel to line W. Segment CF and line V intersect at point H. Find the distance between points D 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 ABCDEFHI be a regular octagon with center G and circumradius 2. Let R be the reflection of point A across point I. Line Q is the perpendicular bisector of segment DR. Line X is the reflection of line Q across point B. Point Z is the reflection of point A across line X. Point U is the reflection of point I across line X. What is the length of segment UZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFG is a regular hexagon with center C and circumradius 1. Point L is the midpoint of segment AB. Point N is the midpoint of segment GL. Let W be the midpoint of segment DN. Find the length of segment BW. 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 M and circumradius 7. Point S is the midpoint of segment BF. Point N is the reflection of point S across segment AE. Point K is the reflection of point C across segment AE. What is the distance between points K and N? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center M and circumradius 3. Let L be the circumcircle of polygon ABCDEFGH. Line N is the perpendicular bisector of segment GE. Let I be the line through point H parallel to line N. Circle Z is the reflection of circle L across line I. Point W is the inversion of point D with respect to circle Z. Let K be the midpoint of segment GD. Find the measure of angle KDW, 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 ACDEFGHI be a regular octagon with center B and circumradius 9. Let O be the circumcircle of polygon ACDEFGHI. Point N is the midpoint of segment IH. Line Z passes through points A and N. Point M is the reflection of point H across line Z. Point Y is the inversion of point N with respect to circle O. Find the length of segment YM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center O and circumradius 7. Point X is the reflection of point O across point B. Let P be the midpoint of segment XJ. Point W is the midpoint of segment CF. Let V be the angle bisector of angle BPW. Point N is the reflection of point B across line V. Find the length of segment NB. 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 M and circumradius 5. Let S be the line through points C and M. Line P is the perpendicular bisector of segment AB. Line V is the reflection of line S across line P. Point Y is the midpoint of segment DA. Point H is on line S with |HD| = 11. Let Q be the midpoint of segment DH. Segment CY and line V intersect at point W. Let U be the reflection of point Q across point D. What is the distance between points W 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 ABCDEFHI be a regular octagon with center G and circumradius 8. Point L is the midpoint of segment GB. Let M be the circumcircle of polygon ABCDEFHI. Let P be the angle bisector of angle LHD. Point V is the reflection of point A across line P. Let Z be the inversion of point V in circle M. What is the distance between points Z 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 ABCDE be a regular 5-gon with center Z and circumradius 6. Let S be the reflection of point Z across point B. Point L is the midpoint of segment DA. Point U is the reflection of point E across point L. Let I be the midpoint of segment SE. What is the distance between points I and U? 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 5. Let X be the circumcircle of polygon ABCDEFGHIJKM. Let O be the reflection of circle X through point E. Line S passes through points M and D. Let Y be the inversion of point L in circle O. Point Z is the reflection of point Y across line S. Find the length of segment FZ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center T and circumradius 3. Point K is the midpoint of segment BT. Let R be the circumcircle of polygon ABCDEFGHIJ. Let U be the inversion of point K in circle R. Find the distance between points U 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 ABCDEF be a regular hexagon with center X and circumradius 6. Let L be the perpendicular bisector of segment DF. Line Q passes through point F parallel to line L. Point S is the reflection of point D across line Q. Let H be the line through points E and S. Lines Q and H intersect at point Z. Point O is the reflection of point Z across line L. Find 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 ABCDEFGH be a regular octagon with center R and circumradius 11. Let J be the vector from point A to point F. Point Q is obtained by rotating point B by angle pi/3 about point H. Point W is the translation of point R by vector J. Line T is the perpendicular bisector of segment BR. Point S is the reflection of point W across line T. Let P be the midpoint of segment SQ. Point V is the reflection of point D across point S. Find the length of segment VP. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center N and circumradius 10. Let T be the reflection of point G across point C. Let L be the line through point G parallel to segment AH. Line V passes through point T perpendicular to line L. Point I is the reflection of point G across line V. Find the distance between points F 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 ABCDFG be a regular hexagon with center E and circumradius 2. Let T be the circumcircle of polygon ABCDFG. Let W be the angle bisector of angle DAE. Point K is the midpoint of segment BC. Let S be the reflection of circle T through point K. Point R is the inversion of point D with respect to circle S. Point U is the reflection of point R across line W. Find the length of segment CU. 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 J and circumradius 7. Line U passes through points G and E. Line S passes through point F perpendicular to line U. Let Y be the line through point F perpendicular to line S. Point Q is the reflection of point B across line Y. What is the length of segment QA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center H and circumradius 2. Line T passes through points E and C. Line N passes through point A perpendicular to line T. Point G is the reflection of point E across line N. Let K be the reflection of point C across point D. Point R is the reflection of point B across segment AK. Find the length of segment RG. 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 12. Line N passes through points G and C. Point J is the reflection of point B across line N. Let R be the line through points C and E. Line W is the reflection of line R across line N. Let Z be the reflection of line W through point A. Line U passes through point A perpendicular to line Z. Point K is the reflection of point J across line U. What is the length of segment BK? 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 S be the midpoint of segment BD. Point J is the midpoint of segment FE. Line Z passes through points W and J. Let K be the line through point A parallel to line Z. Point V is the reflection of point S across line K. Find the distance between points V and B. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center U and circumradius 5. Point G is the midpoint of segment AU. Point Y is obtained by rotating point A by angle 5pi/12 about point U. Point O is the midpoint of segment BE. Line K passes through points C and D. Point I is the reflection of point O across line K. Let M be the midpoint of segment YG. Find the distance between points I and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFG is a regular hexagon with center C and circumradius 4. Let Y be the rotation of point F by angle 3pi/4 around point A. Let X be the angle bisector of angle FYG. Point L is the reflection of point F across line X. Point P is the reflection of point L across segment YF. What is the length of segment YP? 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 6. Let T be the line through points D and B. Point P is the reflection of point K across line T. Let Q be the midpoint of segment JP. Find 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 TG = 9. ABCDEF is a regular hexagon with center G and circumradius 10. Let H be the rotation of point G by angle pi around point D. Point W is the midpoint of segment AG. Point M is the midpoint of segment HW. Line V passes through points G and A. Point Q is the midpoint of segment MF. Point I is the reflection of point Q across line V. What is the length of segment IW? 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 12. Let T be the line through points B and F. Line J passes through point D parallel to line T. Let O be the reflection of line T through point A. Line Q is the reflection of line J across line T. Line R is the reflection of line O across line Q. Line X is the reflection of line O across line R. Line X and segment DH intersect at point P. 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]
ABCDEF is a regular hexagon with center T and circumradius 7. Let P be the rotation of point A by angle 5pi/12 around point E. Point I is the midpoint of segment FT. Point J is the reflection of point A across segment PI. Point R is the reflection of point J across point P. What is the distance between points F and R? 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 T and circumradius 3. Let U be the rotation of point G by angle 3pi/4 around point B. Let O be the circumcircle of polygon ABCDEFGHIJK. Circle P is the reflection of circle O across point F. Line S is the perpendicular bisector of segment TU. Let Y be the reflection of circle P across line S. Let V be the line through points A and I. Circle N is the reflection of circle Y across line V. Let R be the inversion of point A in circle N. What is the length of segment RF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center R and circumradius 4. Point X is the reflection of point F across point A. Let I be the line through points B and H. Point T is the reflection of point C across line I. Point Q is the reflection of point G across point X. What is the measure of angle TGQ, 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 U and circumradius 1. Line Y passes through point F parallel to segment HU. Line K passes through points D and A. Let N be the reflection of line K through point E. Lines Y and N intersect at point I. Find the distance between points I and E. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEGHI be a regular octagon with center F and circumradius 11. Let K be the circumcircle of polygon ABCDEGHI. Line N is the angle bisector of angle GIA. Let T be the angle bisector of angle IBC. Point W is the reflection of point H across line T. Let V be the reflection of circle K across line N. Let L be the inversion of point B in circle V. Find the length of segment WL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center P and circumradius 10. Line M passes through points C and A. Line L is the perpendicular bisector of segment EF. Let J be the reflection of line L through point E. Lines M and J intersect at point O. Find the distance between points O 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 Y and circumradius 10. Point U is obtained by rotating point Y by angle 3pi/4 about point F. Let P be the circumcircle of polygon ABCDEFGH. Let Q be the reflection of point C across point U. Point T is the inversion of point Q with respect to circle P. Let R be the midpoint of segment YT. Point I is the midpoint of segment BR. What is the distance between points F 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 ABCDEFGHIJKL be a regular dodecagon with center Q and circumradius 4. Let M be the circumcircle of polygon ABCDEFGHIJKL. Point P is the reflection of point B across point I. Point V is the reflection of point I across point P. Point U is the inversion of point P with respect to circle M. What is the length of segment UV? 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 K and circumradius 1. Point E is the reflection of point K across segment DC. Let X be the reflection of point E across point C. Let H be the midpoint of segment KD. Find the distance between points H and X. 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 Y and circumradius 4. Point T is the midpoint of segment CY. Let L be the midpoint of segment TB. Point V is the reflection of point C across point D. Let E be the reflection of point L across point A. Point W is the midpoint of segment EV. What is the length of segment WB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center H and circumradius 12. Line J passes through points C and B. Point I is the reflection of point H across line J. Let S be the reflection of point F across point I. Point W is the reflection of point S across line J. What is the distance between points W and C? 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 M and circumradius 2. Let W be the midpoint of segment BA. Let V be the midpoint of segment MW. Let H be the reflection of point A across point V. Point U is the midpoint of segment AV. What is the length of segment UH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center S and circumradius 8. Let J be the angle bisector of angle DBE. Let I be the line through point D perpendicular to line J. Line K passes through point G parallel to line J. Point Z is the midpoint of segment SF. Line T passes through points Z and C. Let P be the reflection of line K across line T. Let R be the reflection of line I through point E. Lines P and R intersect at point L. Point W is the midpoint of segment GL. What is the length of segment SW? 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. Let P be the line through points C and E. Let G be the midpoint of segment BC. Point L is the reflection of point G across line P. Find the distance between points X 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 ABCDE be a regular 5-gon with center M and circumradius 3. Let G be the reflection of point M across point C. Let O be the midpoint of segment BG. Line T passes through points B and D. Point F is the reflection of point O across line T. Let W be the midpoint of segment AM. 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]
ABCDEFGH is a regular octagon with center O and circumradius 1. Line R is the perpendicular bisector of segment BF. Let T be the line through point C parallel to line R. Point Q is the reflection of point F across line T. Let Y be the reflection of point Q across point C. Find the length of segment YQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFGH be a regular 7-gon with center C and circumradius 1. Let J be the circumcircle of polygon ABDEFGH. Let M be the reflection of circle J through point E. Point R is the reflection of point C across point A. Let Y be the inversion of point C in circle M. What is the distance between points Y and R? 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 7. Line Z is the perpendicular bisector of segment HI. Let U be the line through point L parallel to line Z. Point Y is the reflection of point H across line U. Point T is the reflection of point X across point D. What is the distance between points T and Y? 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 3. Let Z be the rotation of point E by angle pi/3 around point B. Let M be the reflection of point F across point C. Let X be the reflection of point A across point Z. Let P be the reflection of point X across point E. Point Y is the reflection of point P across segment MX. Find the distance between points Y and B. 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 4. Line L passes through points D and B. Line Q passes through point A perpendicular to line L. Line H is the reflection of line Q across point D. Let G be the reflection of point D across point A. Line O is the reflection of line H across point B. Let T be the reflection of line H across line O. Point U is the reflection of point G across line T. What is the distance between points U 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 ABCDEFGHI be a regular 9-gon with center V and circumradius 7. Let L be the angle bisector of angle BDH. Line R passes through point E parallel to line L. Line R and segment FA intersect at point M. Let T be the midpoint of segment FA. 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]
Let ABCDEFGHIJ be a regular 10-gon with center O and circumradius 1. Point L is the midpoint of segment IG. Point T is obtained by rotating point A by angle pi/2 about point D. Let M be the circumcircle of polygon ABCDEFGHIJ. Point P is the reflection of point J across segment BA. Let N be the inversion of point T in circle M. Point Y is the midpoint of segment PL. What is the distance between points N and Y? 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 12. Let R be the line through points K and D. Point U is the reflection of point E across line R. Point S is the intersection of line R and segment BX. What is the length of segment US? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YE = 3. Let ABCDFGHIJKL be a regular 11-gon with center E and circumradius 7. Point X is the midpoint of segment FE. Let U be the angle bisector of angle CKJ. Let R be the reflection of line U through point X. Point S is the reflection of point X across line R. What is the distance between points S and D? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFGHI is a regular octagon with center D and circumradius 5. Line U is the perpendicular bisector of segment AI. Let M be the midpoint of segment HI. Let V be the reflection of line U through point E. Segment GB and line V intersect at point X. What is the distance between points X 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 L and circumradius 5. Line O is the angle bisector of angle LDB. Let M be the circumcircle of polygon ABCDEF. Let Z be the line through point B perpendicular to line O. Circle N is the reflection of circle M across line Z. Point Y is the inversion of point E with respect to circle N. Let K be the reflection of point L across point Y. Find the length of segment DK. 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 7. Let X be the line through points F and A. Line T passes through point Q parallel to segment FD. Let P be the reflection of line X through point C. Let H be the midpoint of segment FD. Lines T and P intersect at point K. Find the distance between points K and H. 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 J and circumradius 8. Point P is obtained by rotating point J by angle pi/6 about point H. Let R be the line through points D and A. Line N passes through point G perpendicular to segment PA. Line Z is the reflection of line R across line N. Line S is the reflection of line Z across point F. Point X is the intersection of line S and segment PA. Find the distance between points E 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 EP = 3. Let ABCDFG be a regular hexagon with center P and circumradius 6. Let L be the line through points F and D. Point X is the reflection of point B across line L. Point R is the midpoint of segment XG. Point O is the midpoint of segment DB. Find the measure of angle RDO, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIKL is a regular 11-gon with center J and circumradius 2. Let V be the rotation of point A by angle 5pi/12 around point E. Let Y be the midpoint of segment EF. Line R passes through points Y and I. Line T passes through point G perpendicular to line R. Line O passes through point J perpendicular to line T. Point M is the reflection of point F across line O. Point P is the midpoint of segment MV. What is the length of segment PL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 1. Point W is the reflection of point B across point I. Point N is the reflection of point A across point B. Point O is the reflection of point N across segment WG. What is the distance between points K 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 ABCDEFG be a regular 7-gon with center M and circumradius 6. Line I passes through points E and B. Let O be the circumcircle of polygon ABCDEFG. Let Q be the reflection of circle O across line I. Line R is the reflection of line I across point G. Circle N is the reflection of circle Q across line R. Let H be the inversion of point M in circle N. What is the distance between points F and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHI is a regular 9-gon with center X and circumradius 6. Let N be the reflection of point C across point E. Let T be the angle bisector of angle NBA. Point J is the reflection of point D across point F. Point R is the reflection of point I across line T. What is the distance between points R and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJK is a regular 11-gon with center P and circumradius 12. Point N is the midpoint of segment IK. Point Z is the midpoint of segment AC. Let T be the midpoint of segment NB. Find the length of segment ZT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ACDEF is a regular 5-gon with center B and circumradius 8. Let G be the circumcircle of polygon ACDEF. Let W be the midpoint of segment FB. Let U be the midpoint of segment DB. Let L be the inversion of point W in circle G. Point S is the midpoint of segment UB. What is the distance between points S and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center X and circumradius 6. Line Q passes through points B and D. Point S is the reflection of point X across line Q. Line N passes through points S and A. Point Y is the intersection of lines N and Q. Find 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]
Let ABCDEFGH be a regular octagon with center I and circumradius 10. Line R is the angle bisector of angle FDC. Let P be the reflection of point G across point F. Line V is the angle bisector of angle BPG. Point L is the midpoint of segment BC. Point K is the reflection of point L across line R. Point N is the reflection of point K across line V. Find the length of segment NP. 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 12. Let P be the midpoint of segment FE. Let W be the line through points E and D. Point K is the midpoint of segment CZ. Let X be the reflection of line W through point P. Point Q is the reflection of point D across line X. What is the distance between points K and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center X and circumradius 1. Let I be the circumcircle of polygon ABCDEF. Let T be the reflection of point A across point C. Point O is the inversion of point T with respect to circle I. Point U is the midpoint of segment XB. Find the distance between points U 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 ABDEFGHIJK be a regular 10-gon with center C and circumradius 6. Line U passes through points J and C. Let O be the line through points B and D. Point Y is the intersection of lines O and U. Point T is the midpoint of segment AY. What is the distance between points C and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center K and circumradius 7. Let Q be the midpoint of segment HF. Point P is the midpoint of segment KQ. Point Y is the reflection of point P across segment FK. Find the length of segment YQ. 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 5. Point M is the midpoint of segment SB. Line I is the perpendicular bisector of segment MF. Point L is the reflection of point E across segment AF. Point X is the reflection of point L across line I. What is the distance between points S and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 5. Let V be the midpoint of segment BF. Let X be the midpoint of segment DV. Line R is the perpendicular bisector of segment LJ. Point O is the reflection of point X across line R. Find the distance between points O 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 ABCDE be a regular 5-gon with center G and circumradius 9. Line N is the angle bisector of angle GDB. Point O is the reflection of point G across line N. Line F is the angle bisector of angle CAO. Line H passes through point D parallel to line F. Let L be the midpoint of segment BD. Point T is the reflection of point O across line H. Point S is the reflection of point L across point C. Point K is the reflection of point T across line N. What is 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 BCDEFG be a regular hexagon with center A and circumradius 4. Point X is the reflection of point E across point F. Let S be the midpoint of segment EA. Let N be the midpoint of segment XS. What is the length of segment ND? 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 2. Let S be the reflection of point D across point L. Let Q be the midpoint of segment SL. Point R is the reflection of point H across point K. What is the length of segment RQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEGHIJ is a regular 9-gon with center F and circumradius 10. Let T be the reflection of point B across point I. Line L is the angle bisector of angle CGD. Let N be the reflection of point F across point T. Point Z is the reflection of point N across line L. Find the distance between points Z and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center K and circumradius 9. Let H be the reflection of point E across point C. Let L be the line through point H parallel to segment AF. Point R is the reflection of point D across line L. Point T is the midpoint of segment AF. Point Y is the reflection of point T across segment RD. Find the measure of angle KAY, 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 ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 6. Line O is the angle bisector of angle KCE. Let N be the line through point Y parallel to line O. Let U be the reflection of line N across line O. Let Z be the angle bisector of angle BFJ. Let W be the reflection of line Z across line U. Let P be the reflection of line W through point A. Point T is the reflection of point G across line P. Let M be the midpoint of segment HT. What is 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 BCDEFGHIJKLM be a regular dodecagon with center A and circumradius 3. Point U is the midpoint of segment LE. Point Q is the reflection of point B across segment LE. Let T be the reflection of point G across point Q. What is the length of segment UT? 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 8. Line S passes through points B and E. Line S and segment GD intersect at point Y. Point W is the midpoint of segment AY. 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 K and circumradius 8. Line O passes through points A and D. Point M is the reflection of point B across point D. Let Z be the line through points E and D. Let J be the reflection of line Z across line O. Point Y is the reflection of point M across line J. Find the distance between points Y 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 ABCDE be a regular 5-gon with center T and circumradius 5. Point S is obtained by rotating point C by angle 3pi/4 about point B. Point F is the midpoint of segment CE. Let M be the midpoint of segment DB. Point G is the reflection of point F across segment DS. Find 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]
ABCDE is a regular 5-gon with center J and circumradius 2. Point M is the reflection of point A across point E. Point G is the intersection of segments MJ and AD. Point V is the reflection of point A across segment MJ. Find 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]
ABCDE is a regular 5-gon with center Z and circumradius 12. Point I is obtained by rotating point A by angle 3pi/4 about point B. Let V be the line through points Z and B. Line P passes through point I parallel to line V. Let H be the reflection of line P across line V. Point T is the intersection of line H and segment ZA. What is the distance between points T 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 ABCDEFGHIJ be a regular 10-gon with center K and circumradius 1. Let L be the circumcircle of polygon ABCDEFGHIJ. Let V be the angle bisector of angle FAJ. Point P is the reflection of point C across line V. Point X is the inversion of point P with respect to circle L. Find the distance between points X and J. Express your answer as a decimal number rounded to 2 decimal places.

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


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

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


[User]
Let ABCDEFGH be a regular octagon with center K and circumradius 11. Let X be the midpoint of segment DG. Point Z is the reflection of point X across point K. Point U is the midpoint of segment FZ. What is the distance between points B and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEFGHI be a regular octagon with center D and circumradius 11. Let X be the reflection of point I across point H. Point L is the midpoint of segment HX. Let K be the reflection of point A across point H. Find the length of segment KL. 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 R and circumradius 8. Let Z be the circumcircle of polygon ABCDEFGHI. Let X be the reflection of circle Z through point G. Let Q be the midpoint of segment DE. Point M is the midpoint of segment CQ. Point Y is the inversion of point M with respect to circle X. Find the distance between points E and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center Z and circumradius 6. Let P be the reflection of point Z across point A. Let S be the rotation of point P by angle pi/4 around point C. Point O is the midpoint of segment EP. Let M be the perpendicular bisector of segment ZS. Line K passes through point C perpendicular to line M. Point J is the reflection of point O across line K. Find the length of segment JG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center Q and circumradius 3. Let T be the reflection of point D across point C. Let J be the midpoint of segment QT. Let X be the reflection of point A across point J. What is the distance between points X and B? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center X and circumradius 7. Let Q be the line through points F and D. Point K is the reflection of point C across segment FB. Point H is the reflection of point C across line Q. What is the measure of angle KCH, 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 W and circumradius 1. Let T be the perpendicular bisector of segment BW. Let K be the midpoint of segment FD. Line X passes through point G perpendicular to segment FD. Let P be the reflection of line X across line T. Line I is the perpendicular bisector of segment CG. Let O be the reflection of line I across line X. Lines O and P intersect at point Z. Point U is the reflection of point K across line T. What is the distance between points U 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 I and circumradius 6. Let W be the reflection of point A across point B. Point J is the midpoint of segment FB. Point Y is the reflection of point W across point F. Point V is the midpoint of segment JY. Point K is the midpoint of segment AE. What is the length of segment KV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center M and circumradius 8. Point T is the reflection of point D across point G. Line W is the angle bisector of angle TFG. Let U be the reflection of point A across point B. Point Y is the reflection of point U across line W. Find the distance between points Y and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center Q and circumradius 3. Point S is the midpoint of segment HQ. Line J is the perpendicular bisector of segment SG. Point Y is the reflection of point Q across line J. What is the length of segment HY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center Y and circumradius 11. Let V be the midpoint of segment CD. Let Q be the midpoint of segment FB. Point U is the reflection of point B across segment VD. What is the measure of angle UQV, 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 M and circumradius 11. Let L be the line through point C perpendicular to segment AC. Point T is the reflection of point F across line L. Let S be the reflection of point T across point A. Point Z is the reflection of point S across segment AC. What is the distance between points Z 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 ABCDEFHIJK be a regular 10-gon with center G and circumradius 10. Point L is the reflection of point B across point F. Let Y be the reflection of point B across point L. Let V be the midpoint of segment YI. Find the distance between points B and V. 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 6. Let S be the rotation of point F by angle pi/12 around point B. Let K be the perpendicular bisector of segment DF. Line L passes through point F parallel to line K. Let U be the perpendicular bisector of segment HS. Let J be the reflection of line U across line L. Line J and segment HS intersect at point O. What is 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 ABCDEG be a regular hexagon with center F and circumradius 2. Point H is the reflection of point A across point C. Point Q is the midpoint of segment GH. Let O be the reflection of point Q across point C. Point I is the midpoint of segment OB. What is the length of segment ID? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BCDEF be a regular 5-gon with center A and circumradius 11. Let M be the reflection of point D across point C. Let X be the circumcircle of polygon BCDEF. Point L is the reflection of point F across point M. Point K is the inversion of point L with respect to circle X. Find 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]
BN = 10. Let ACDEFG be a regular hexagon with center N and circumradius 7. Let S be the reflection of point E across point C. Line L is the perpendicular bisector of segment DS. Let O be the line through points C and G. Point I is the reflection of point G across line L. Point J is the reflection of point I across line O. What is the length of segment JA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center O and circumradius 7. Point Y is the intersection of segments FK and OH. Point R is the reflection of point Y across point K. Point Z is the reflection of point E across point B. What is the length of segment ZR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFGHIJK be a regular 10-gon with center B and circumradius 11. Let O be the line through points G and D. Line L passes through point A perpendicular to line O. Point M is the reflection of point I across point F. Point U is the reflection of point J across line L. Point V is the midpoint of segment UM. What is the distance between points K and V? 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 R and circumradius 7. Point M is the midpoint of segment CR. Let T be the circumcircle of polygon ABCDEFGH. Let J be the inversion of point M in circle T. What is the distance between points J and A? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center U and circumradius 7. Point N is obtained by rotating point U by angle pi/12 about point A. Line M is the angle bisector of angle DAE. Line L passes through points N and E. Line F is the reflection of line L across line M. Point S is the reflection of point N across line F. Find the length of segment SU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let LZ = 3. Let ABCDEFGHIJKM be a regular dodecagon with center Z and circumradius 4. Let R be the line through point D parallel to segment HZ. Point S is on line R. Point P is the reflection of point S across segment HZ. Find the distance between points P and S. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center Q and circumradius 6. Let W be the line through points A and D. Line P is the reflection of line W across point B. Point S is the reflection of point C across line P. Let T be the midpoint of segment SD. Find the distance between points T and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center S and circumradius 5. Let I be the circumcircle of polygon ABCDEF. Let H be the line through points A and C. Let M be the midpoint of segment ES. Let L be the reflection of circle I through point D. Let Q be the inversion of point A in circle L. Point R is the reflection of point Q across line H. Find the length of segment MR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIK be a regular 10-gon with center J and circumradius 4. Line O is the perpendicular bisector of segment DK. Line T passes through point H parallel to line O. Point P is the midpoint of segment KF. Point V is the reflection of point P across line T. What is the length of segment DV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 6. Let S be the circumcircle of polygon ABDEFGHIJKLM. Let Z be the line through points K and D. Let V be the midpoint of segment LG. Line Y is the reflection of line Z across point C. Circle R is the reflection of circle S across line Y. Point U is the reflection of point A across line Y. Let T be the midpoint of segment DU. Let O be the inversion of point T in circle R. What is the distance between points V and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center L and circumradius 11. Let Z be the circumcircle of polygon ABCDE. Let K be the perpendicular bisector of segment BL. Point R is the reflection of point C across line K. Point P is the inversion of point R with respect to circle Z. What is the length of segment EP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 5. Let Z be the reflection of point G across point J. Point Y is the midpoint of segment MC. Point W is the midpoint of segment HY. What is the distance between points W and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center M and circumradius 1. Point Q is the reflection of point E across segment BA. Let X be the reflection of point G across point C. Line V is the angle bisector of angle HEQ. Point O is the reflection of point H across line V. Let P be the midpoint of segment XM. 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]
Let ABDE be a square with center C and circumradius 5. Line V passes through points E and D. Let T be the circumcircle of polygon ABDE. Let M be the reflection of circle T across line V. Point J is the inversion of point A with respect to circle M. What is the measure of angle AJE, 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 ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 10. Let O be the reflection of point G across point J. Point U is the midpoint of segment JB. Point Y is the midpoint of segment HO. Find the length of segment UY. 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 11. Let M be the line through point E parallel to segment AD. Let V be the reflection of point G across point F. Point W is the reflection of point V across line M. What is the distance between points N and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGIJK be a regular 10-gon with center H and circumradius 10. Let U be the midpoint of segment EB. Let V be the perpendicular bisector of segment IE. Point P is the reflection of point D across point U. Point W is the reflection of point P across line V. What is the measure of angle JDW, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 10. Line Y is the perpendicular bisector of segment KF. Let W be the angle bisector of angle AHB. Let V be the reflection of line Y across line W. Line U passes through points A and D. Let N be the reflection of line U across line V. Point Z is the reflection of point G across line N. Find the length of segment EZ. 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 U be the line through points F and L. Let Y be the perpendicular bisector of segment EK. Let X be the reflection of line Y through point J. Line V is the reflection of line X across line U. Line M passes through point Z parallel to segment IJ. Let O be the reflection of line M across line U. Lines O and V intersect at point R. What is the distance between points R 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 ABCDE be a regular 5-gon with center G and circumradius 9. Let U be the perpendicular bisector of segment EG. Point W lies on line U at distance 7 from point C. Line F passes through point B perpendicular to segment GE. Point T is the reflection of point W across segment GE. Point V is the reflection of point T across line F. What is the distance between points T and V? 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 R and circumradius 5. Line N is the angle bisector of angle ABC. Let F be the reflection of line N through point C. Let W be the line through points C and A. Point G is the intersection of lines F and W. Let X be the midpoint of segment GD. Find the distance between points A and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center X and circumradius 1. Let Q be the reflection of point E across point A. Line N passes through points Q and H. Line S passes through point H perpendicular to line N. Point L is the reflection of point E across line S. What is the distance between points D 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 ABCDEFG be a regular 7-gon with center Z and circumradius 2. Point X is the midpoint of segment AF. Line Y is the perpendicular bisector of segment CZ. Line N is the reflection of line Y across point X. Point U is the reflection of point E across line N. 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 N and circumradius 9. Let Y be the line through points G and L. Point X is the intersection of line Y and segment IA. Point O is the reflection of point L across segment IA. What is the length of segment OX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center T and circumradius 6. Let R be the midpoint of segment CA. Point Z is the midpoint of segment ER. Point J is the midpoint of segment BZ. What is the distance between points J and T? 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 5. Point W is the midpoint of segment GF. Line S is the perpendicular bisector of segment DW. Let Z be the line through point P perpendicular to line S. Point Y is the reflection of point F across line Z. Point X is the reflection of point G across line Z. Let Q be the reflection of point X across point A. Find the distance between points Y and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFHIJKL is a regular 11-gon with center G and circumradius 2. Let S be the circumcircle of polygon ABCDEFHIJKL. Let O be the angle bisector of angle KDL. Point T is the reflection of point A across line O. Let R be the reflection of circle S through point J. Let U be the inversion of point F in circle R. What is the length of segment UT? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center N and circumradius 9. Let Z be the line through points N and A. Line X is the angle bisector of angle DEA. Let L be the reflection of line X across line Z. Point P is the reflection of point B across line L. 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]
Let ABCDEF be a regular hexagon with center S and circumradius 4. Let R be the rotation of point F by angle pi/4 around point S. Point Q is the reflection of point B across point F. Let O be the line through points R and Q. Let N be the angle bisector of angle RSC. Let H be the reflection of line N across line O. Point P is the reflection of point E across line H. What is the distance between points P and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJ be a regular 10-gon with center U and circumradius 11. Point K is the midpoint of segment EB. Segments KA and CJ intersect at point O. Find the distance between points K and O. 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 T and circumradius 8. Line M is the perpendicular bisector of segment AT. Let N be the reflection of point D across point F. Point V is obtained by rotating point T by angle 5pi/6 about point F. Line S passes through point N perpendicular to line M. Point L is on line S. Let W be the angle bisector of angle LVF. Point P is the reflection of point T across line W. Find the length of segment PV. 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 X and circumradius 1. Let R be the circumcircle of polygon ABCDEFGHI. Let M be the midpoint of segment HI. Point T is the inversion of point M with respect to circle R. What is the distance between points C and T? 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 9. Line S is the angle bisector of angle YDC. Let U be the midpoint of segment AY. Let O be the reflection of line S through point F. Point Q is the reflection of point U across line O. Find the length of segment QG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center Q and circumradius 9. Point Y is obtained by rotating point J by angle pi/6 about point I. Let O be the angle bisector of angle GIY. Line M passes through point B perpendicular to line O. Let V be the reflection of line M through point L. Let N be the reflection of line V through point K. Point U is the reflection of point B across line N. What is the distance between points U and D? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGI be a regular octagon with center H and circumradius 1. Line L passes through points I and H. Point V is the reflection of point I across point E. Let X be the perpendicular bisector of segment GE. Line W passes through point V perpendicular to line L. Line U is the reflection of line X across line W. Point R is the reflection of point A across line U. What is the length of segment AR? 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 9. Let L be the circumcircle of polygon ABCDEFGH. Point Z is the reflection of point O across point F. Let N be the inversion of point Z in circle L. Point M is the midpoint of segment OE. Point W is the midpoint of segment ME. Let K be the perpendicular bisector of segment OE. Point V is the reflection of point W across line K. What is the distance between points N and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJK be a regular 11-gon with center M and circumradius 1. Point T is the midpoint of segment KB. Point W is the midpoint of segment DE. Line Q is the angle bisector of angle WTJ. Point P is the reflection of point I across line Q. What is the length of segment AP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDFGHIJKLM is a regular dodecagon with center E and circumradius 1. Point T is the midpoint of segment GI. Let W be the angle bisector of angle GEC. Line P is the angle bisector of angle BAG. Point Y is the reflection of point B across line P. Point Q is the reflection of point T across line W. Find the length of segment YQ. 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 I and circumradius 12. Point J is obtained by rotating point D by angle pi/6 about point B. Let R be the line through points J and A. Line Z is the reflection of line R across point B. Point V is the reflection of point B across line R. Point Q is the reflection of point V across line Z. Point K is the midpoint of segment VQ. Find the measure of angle KBC, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center M and circumradius 8. Line O is the perpendicular bisector of segment ME. Point H is the midpoint of segment BM. Point G is the reflection of point H across line O. Find 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]
ABCDEG is a regular hexagon with center F and circumradius 1. Let T be the reflection of point E across point C. Let K be the reflection of point C across point B. Line M is the angle bisector of angle CEF. Point Z is the midpoint of segment AB. Point Y is the reflection of point Z across line M. Let U be the reflection of point Y across point T. What is the length of segment UK? 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 6. Let K be the rotation of point E by angle pi/12 around point C. Let X be the vector from point E to point B. Let Y be point C translated by vector X. Point I is the reflection of point K across point M. Let R be the line through points Y and I. Point T is the reflection of point B across line R. Point J is the reflection of point C across segment GD. Find the length of segment JT. 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 Z is the midpoint of segment KF. Let P be the reflection of point I across point Z. Let R be the line through point P parallel to segment AG. Point X is the reflection of point J across line R. Let Y be the reflection of point Z across point G. What is the distance between points X and Y? 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 2. Let X be the perpendicular bisector of segment FA. Line Z is the reflection of line X across point F. Point U is the reflection of point G across line Z. Find the distance between points U 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 12. Point Y is the midpoint of segment ED. Let X be the angle bisector of angle HEC. Let N be the line through point Y parallel to line X. Point Q is the reflection of point O across line N. Find the length of segment JQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center X and circumradius 9. Let J be the reflection of point X across point C. Point G is the midpoint of segment DE. Let L be the midpoint of segment GE. What is the distance between points L and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center R and circumradius 2. Let N be the perpendicular bisector of segment DA. Line O passes through point C perpendicular to line N. Point H is the reflection of point R across line O. Line P is the angle bisector of angle RHD. Point Z is the reflection of point R across line P. Find the distance between points Z and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center U and circumradius 5. Let S be the midpoint of segment IU. Let Z be the line through points C and J. Point Q is the midpoint of segment HD. Let V be the line through points Q and F. Let M be the reflection of line Z through point F. Lines V and M intersect at point O. What is the length of segment OS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center W and circumradius 4. Line G passes through point C perpendicular to segment AB. Let V be the circumcircle of polygon ABCDE. Circle K is the reflection of circle V across line G. Let T be the inversion of point B in circle K. Point N is the midpoint of segment TA. Find the distance between points E and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VA = 8. BCDEFG is a regular hexagon with center A and circumradius 3. Line P is the angle bisector of angle AEB. Line Q is the reflection of line P across point C. Line J is the reflection of line Q across point G. Point H is the reflection of point F across line J. Find the distance between points H 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 ABCDEFG be a regular 7-gon with center Y and circumradius 8. Let X be the midpoint of segment FA. Line I passes through point X parallel to segment GF. Point O lies on line I. Find the measure of angle OXA, 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 2. Let X be the circumcircle of polygon ABCEFGHIJKLM. Circle R is the reflection of circle X across point M. Circle T is the reflection of circle R across point F. Point W is the inversion of point G with respect to circle T. What is 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]
ABCDEGHIJK is a regular 10-gon with center F and circumradius 9. Let P be the perpendicular bisector of segment IE. Let L be the reflection of line P through point A. Point X is the reflection of point G across line L. Let M be the reflection of point G across point X. Let Z be the reflection of point M across point B. Find the distance between points Z 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 ABCDEFG be a regular 7-gon with center W and circumradius 3. Line M is the angle bisector of angle GDW. Let O be the line through points W and D. Line P is the reflection of line O across line M. Point T is the reflection of point C across line P. What is the distance between points B 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 ABCDEFGHIJK be a regular 11-gon with center R and circumradius 2. Let X be the midpoint of segment AD. Let S be the line through points I and D. Line T passes through point X parallel to line S. Point O is the reflection of point A across line T. Let N be the reflection of point O across point R. Find the length of segment BN. 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 O be the circumcircle of polygon ABCDEFGHIJ. Let W be the reflection of circle O through point G. Circle P is the reflection of circle W across point V. Let L be the inversion of point D in circle P. Point R is the midpoint of segment CB. Point X is the midpoint of segment LA. Let Y be the midpoint of segment RX. Find the length of segment BY. 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 K and circumradius 9. Point T is the reflection of point K across segment AC. Line N passes through point T perpendicular to segment AC. Let I be the midpoint of segment KA. Point Y is the reflection of point I across segment AC. Point H is the reflection of point I across line N. Find the distance between points H 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 H and circumradius 6. Let O be the angle bisector of angle BEC. Let Z be the rotation of point C by angle pi/4 around point F. Let G be the angle bisector of angle DZC. Line R is the reflection of line G across line O. Line L is the reflection of line O across line R. Point S is the intersection of lines R and L. Point K is the reflection of point Z across point S. Point Y is the reflection of point K across line R. Find the distance between points A and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHI is a regular 9-gon with center M and circumradius 9. Let X be the midpoint of segment FM. Point T is the midpoint of segment AG. Let W be the midpoint of segment DT. Find the distance between points X and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center X and circumradius 1. Line M is the angle bisector of angle CBE. Let Y be the line through point A parallel to line M. Let Z be the reflection of line Y through point D. Point H is the reflection of point C across line Z. What is the distance between points E and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center Z and circumradius 10. Line X passes through points A and Z. Line X and segment DB intersect at point U. Point Q is the reflection of point C across segment DB. Let M be the reflection of point E across point B. Find the measure of angle UQM, 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 S and circumradius 9. Point J is the midpoint of segment BE. Let O be the midpoint of segment JE. Point V is the midpoint of segment BO. Point Z is the reflection of point V across segment DB. 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]
Let ABCDEF be a regular hexagon with center I and circumradius 7. Let J be the perpendicular bisector of segment IA. Point R is the midpoint of segment BD. Line U passes through point I perpendicular to line J. Point M is the reflection of point R across line U. Find the distance between points E and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFGHIJK is a regular 10-gon with center A and circumradius 11. Let Z be the midpoint of segment KE. Let W be the line through points C and J. Let M be the line through point Z parallel to line W. Line S passes through point K parallel to line M. Point O is the reflection of point Z across segment GC. Point T is the reflection of point O across line S. What is 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 ABCDEFGHJKL be a regular 11-gon with center I and circumradius 1. Line S passes through points L and I. Point M is the reflection of point I across point J. Line O passes through point M perpendicular to line S. Point W is the reflection of point G across line O. What is the length of segment WK? 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 11. Line H is the perpendicular bisector of segment ED. Point S is the midpoint of segment EB. Line Z passes through point D perpendicular to line H. Point Q is the reflection of point S across line Z. Find the length of segment EQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center W and circumradius 5. Point U is the midpoint of segment GE. Point X is the reflection of point D across segment HU. Point Z is the reflection of point A across segment HU. What is the distance between points Z and X? 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 7. Line O passes through points N and C. Let S be the midpoint of segment AB. Let L be the midpoint of segment GA. Point I is the reflection of point S across line O. What is the length of segment IL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 11. Point T is the reflection of point K across point E. Point Q is the reflection of point A across segment BK. Let S be the perpendicular bisector of segment MH. Let N be the reflection of line S through point T. Line R passes through points A and Q. Lines R and N intersect at point U. Find the length of segment UK. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDFGHIJKL be a regular 11-gon with center E and circumradius 8. Let U be the circumcircle of polygon ABCDFGHIJKL. Circle W is the reflection of circle U across point H. Let N be the reflection of circle W through point D. Let M be the reflection of circle N through point G. Let T be the inversion of point B in circle M. What is the length of segment TF? 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 W and circumradius 5. Line S passes through points J and C. Line T passes through point F parallel to line S. Line O passes through points A and K. Point Y is the intersection of lines O and T. Find the distance between points Y and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center O and circumradius 12. Let I be the midpoint of segment OE. Let Q be the line through points B and I. Let H be the line through point F perpendicular to line Q. Let U be the reflection of point E across point C. Let N be the line through points U and A. Let L be the reflection of line H across line N. Lines N and L intersect at point V. What is the measure of angle VDB, 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 O and circumradius 7. Let U be the reflection of point G across point F. Line Y passes through point U perpendicular to segment OH. Let V be the circumcircle of polygon ABCDEFGH. Circle X is the reflection of circle V across line Y. Line L passes through points H and B. Circle Q is the reflection of circle X across line L. Point M is the inversion of point D with respect to circle Q. Point K is the midpoint of segment OH. Find the distance between points M and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFG be a regular 7-gon with center Y and circumradius 11. Line Q passes through point G parallel to segment YC. Line V passes through points C and B. Lines V and Q intersect at point N. Point P is the midpoint of segment FN. Find 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]
ABCDEF is a regular hexagon with center Z and circumradius 10. Let N be the circumcircle of polygon ABCDEF. Let X be the midpoint of segment BZ. Let H be the inversion of point X in circle N. Find the length of segment HD. 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 10. Point R is the reflection of point H across segment GL. Point O is the intersection of segments RI and GL. Point Q is the midpoint of segment FI. 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 ABCDEFGHIJKL be a regular dodecagon with center W and circumradius 12. Point U is the reflection of point A across point E. Let X be the midpoint of segment KW. Point O is obtained by rotating point X by angle 7pi/12 about point F. Line S passes through point C parallel to segment KO. Line S and segment BU intersect at point V. What is the distance between points V and D? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center X and circumradius 1. Let R be the reflection of point A across point E. Let Z be the midpoint of segment XA. Point U is the reflection of point B across point X. Point N is the midpoint of segment ZR. Let Q be the midpoint of segment UN. Find the length of segment UQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center K and circumradius 2. Let I be the angle bisector of angle HCE. Line U passes through point D parallel to line I. Let R be the reflection of line U through point E. Point Z is the reflection of point C across line R. What is the distance between points Z and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center V and circumradius 5. Let U be the perpendicular bisector of segment HF. Line S passes through point A parallel to line U. Point Y is the reflection of point H across line S. Let Q be the reflection of point B across point Y. Point X is the reflection of point E across line S. Point T is the midpoint of segment BX. Find the length of segment QT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center N and circumradius 3. Let V be the line through points B and E. Let M be the angle bisector of angle DEG. Line I is the reflection of line M across line V. Line T passes through points D and G. Point Z is the intersection of lines T and I. Find the distance between points F and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BCDEFG be a regular hexagon with center A and circumradius 8. Let X be the reflection of point G across point C. Line I passes through points X and F. Point K is the reflection of point G across point F. Point J is the reflection of point F across point C. Line S passes through point K perpendicular to segment JK. Point L is the reflection of point E across line S. Line I and segment JK intersect at point O. What is the length of segment OL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center W and circumradius 12. Let J be the midpoint of segment WE. Point U is the reflection of point D across point W. Point O is the midpoint of segment UJ. Let K be the perpendicular bisector of segment CO. Point R is the reflection of point B across line K. What is the length of segment JR? 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 12. Point M is the midpoint of segment AH. Let N be the midpoint of segment JF. Point Y is the reflection of point M across segment GN. What is the length of segment BY? 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 1. Let E be the circumcircle of polygon ABCD. Line U is the angle bisector of angle GCB. Point P is the reflection of point A across line U. Let N be the line through point P parallel to line U. Circle I is the reflection of circle E across line N. Point T is the reflection of point G across point C. Let O be the inversion of point T in circle I. 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]
ABCEFGHIJKL is a regular 11-gon with center D and circumradius 3. Point T is the reflection of point K across segment JG. Let N be the perpendicular bisector of segment TB. Let Z be the line through point C parallel to segment KF. Point R is the reflection of point J across line Z. Point Y is the reflection of point J across line N. Find the distance between points R 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 ABCDFGHI be a regular octagon with center E and circumradius 3. Line Q passes through points G and A. Line Z passes through point B parallel to segment IA. Point M is the intersection of lines Z and Q. What is the measure of angle ADM, 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 Y and circumradius 9. Let R be the line through points A and E. Line T is the reflection of line R across point D. Point G is the reflection of point F across line T. Point K is the midpoint of segment YA. Point X is the reflection of point C across point K. Point S is the reflection of point E across segment GA. Find the length of segment SX. 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 7. Let O be the reflection of point B across point L. Point V is the midpoint of segment KJ. Point W is the reflection of point O across segment DG. Find the distance between points W and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center X and circumradius 7. Let R be the circumcircle of polygon ABCDEFGH. Line K passes through points B and H. Point I is the reflection of point A across line K. Point S is the inversion of point I with respect to circle R. Find the distance between points S and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFG is a regular hexagon with center A and circumradius 7. Point N is the midpoint of segment GB. Let S be the reflection of point N across point F. Line X passes through points G and E. Point R is the reflection of point S across line X. What is the length of segment RE? Express your answer as a decimal number rounded to 2 decimal places.

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