[User]
Let ABCDEFGHJKLM be a regular dodecagon with center I and circumradius 3. Let Z be the circumcircle of polygon ABCDEFGHJKLM. Let R be the reflection of circle Z through point H. Line Q passes through points F and K. Circle U is the reflection of circle R across line Q. Point P is the reflection of point G across point H. Line W is the perpendicular bisector of segment KM. Point Y is the reflection of point P across line W. Let S be the inversion of point Y in circle U. Find the distance between points H 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 ABCDFGH be a regular 7-gon with center E and circumradius 11. Line Q passes through point F parallel to segment AH. Point I is the reflection of point B across line Q. Line O is the perpendicular bisector of segment AH. Point Z is the reflection of point I across line O. Find the measure of angle HZC, 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 ABCDEFG be a regular 7-gon with center Z and circumradius 3. Let Y be the circumcircle of polygon ABCDEFG. Line R is the perpendicular bisector of segment ZG. Circle L is the reflection of circle Y across line R. Point P is the inversion of point D with respect to circle L. Let Q be the midpoint of segment FZ. Find the length of segment QP. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center J and circumradius 10. Let Q be the midpoint of segment EF. Let Y be the midpoint of segment QB. Let K be the line through points J and C. Let O be the angle bisector of angle JDE. Let I be the reflection of line O through point C. Point S is the intersection of lines I and K. What is the distance between points S and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center W and circumradius 2. Let Z be the reflection of point A across point E. Let L be the line through points F and D. Point X is the reflection of point Z across line L. Let Q be the reflection of point X across point A. Find the measure of angle EZQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center Y and circumradius 8. Let Z be the midpoint of segment BC. Line S passes through points Z and B. Let T be the reflection of point Z across point C. Let E be the reflection of line S through point Y. Let J be the circumcircle of polygon ABCD. Point P is the midpoint of segment BT. Let I be the reflection of circle J across line S. Point O is the reflection of point Y across point B. Let M be the reflection of line E through point P. Circle V is the reflection of circle I across line M. Let G be the inversion of point O in circle V. Find the length of segment GC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center F and circumradius 1. Let N be the angle bisector of angle AFE. Let Q be the line through point A parallel to line N. Let V be the reflection of line N across line Q. Let Y be the midpoint of segment EF. Point W is the reflection of point Y across point F. Point R is the reflection of point W across line V. What is 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]
ABCDEF is a regular hexagon with center I and circumradius 10. Point R is obtained by rotating point I by angle 5pi/12 about point D. Let W be the circumcircle of polygon ABCDEF. Line G passes through points I and F. Point V is the reflection of point R across line G. Point N is the inversion of point V with respect to circle W. What is the distance between points V 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 ACDEF be a regular 5-gon with center B and circumradius 5. Point Q is the midpoint of segment AC. Let H be the perpendicular bisector of segment DQ. Line M is the reflection of line H across point E. Point J is the reflection of point F across line M. What is the length of segment JC? 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 N and circumradius 3. Let K be the reflection of point F across point A. Line L passes through point A perpendicular to segment KC. Point M is the reflection of point G across line L. Find the distance between points M and D. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFGHI be a regular octagon with center B and circumradius 7. Let M be the rotation of point H by angle 5pi/6 around point C. Let K be the midpoint of segment CH. Line P is the perpendicular bisector of segment IM. Point L is the reflection of point D across line P. 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]
ABCDEFGHIJK is a regular 11-gon with center Q and circumradius 3. Let W be the angle bisector of angle EAK. Let V be the reflection of line W through point C. Line O passes through point B parallel to line V. Let X be the reflection of line V across line O. Point M is the reflection of point Q across line X. Let R be the reflection of line X through point M. Point P is the reflection of point A across line V. Point Z is the reflection of point D across line R. Find the distance between points Z and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 4. Let N be the circumcircle of polygon ABCDEFGIJKLM. Let P be the midpoint of segment ME. Let Q be the angle bisector of angle JEI. Let R be the reflection of line Q through point P. Circle T is the reflection of circle N across point D. Circle Z is the reflection of circle T across line R. Let Y be the inversion of point K in circle Z. What is the length of segment YF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFHIJK be a regular 10-gon with center G and circumradius 2. Let U be the midpoint of segment FH. Let X be the angle bisector of angle BDU. Point L is the reflection of point K across line X. Let R be the circumcircle of polygon ABCDEFHIJK. Point O is the inversion of point L with respect to circle R. What is the distance between points O and C? 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 3. Point J is the intersection of segments EB and DG. Let H be the reflection of point F across point G. Let M be the midpoint of segment JH. What is the distance between points M and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHI is a regular 9-gon with center O and circumradius 8. Let N be the circumcircle of polygon ABCDEFGHI. Let K be the reflection of point D across point G. Point M is the reflection of point B across point K. Point X is the inversion of point M with respect to circle N. What is the distance between points X and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center R and circumradius 11. Line Y passes through points B and L. Let N be the circumcircle of polygon ABCDEFGHIJKL. Line S passes through point H parallel to line Y. Let P be the reflection of circle N across line S. Let U be the inversion of point J in circle P. What is the distance between points C and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center P and circumradius 4. Point N is obtained by rotating point P by angle pi/12 about point A. Point G is the midpoint of segment DN. Point T is the reflection of point E across point G. Point I is the reflection of point C across point B. Point M is the reflection of point I across point P. Find the length of segment MT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center Z and circumradius 11. Let M be the reflection of point A across point C. Point I is the midpoint of segment DM. Point Q is obtained by rotating point M by angle pi/4 about point I. Let N be the rotation of point M by angle 2pi/3 around point Q. Let L be the line through point C perpendicular to segment MI. Let F be the line through point N parallel to segment MI. Line T passes through point Q parallel to line L. Lines F and T intersect at point R. Find the distance between points Z and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ACDEFG is a regular hexagon with center B and circumradius 3. Point V is the reflection of point A across point C. Let O be the reflection of point E across point F. Point X is the midpoint of segment GO. Point N is the midpoint of segment DV. Find the distance between points N and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center V and circumradius 5. Let P be the line through points A and H. Let Y be the line through point B perpendicular to line P. Point I is the reflection of point B across line P. Let R be the line through point H parallel to line Y. Point L is the reflection of point I across line R. Point N is the reflection of point L across line P. Find the distance between points H and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center M and circumradius 1. Line Q is the angle bisector of angle AFH. Point O is the reflection of point C across line Q. Point Y is the reflection of point E across segment BO. Find the distance between points M 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 L and circumradius 4. Let Y be the circumcircle of polygon ABCDEF. Let R be the midpoint of segment LF. Line Q is the angle bisector of angle LRD. Let J be the reflection of point L across point B. Circle W is the reflection of circle Y across line Q. Point X is the inversion of point L with respect to circle W. Point T is the reflection of point X across point J. Find the distance between points T and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center L and circumradius 5. Let Z be the circumcircle of polygon ABCDEFGH. Let V be the midpoint of segment DG. Point X is the inversion of point V with respect to circle Z. Line J is the angle bisector of angle LGE. Point O is the reflection of point F across line J. 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]
ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 6. Point S is the midpoint of segment MK. Point Y is the reflection of point I across point B. Point P is the reflection of point J across segment SA. Find the length of segment PY. 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 4. Let Y be the midpoint of segment FI. Point N is the reflection of point H across point D. Let M be the line through points Y and K. Let U be the midpoint of segment GH. Point Z is the reflection of point N across line M. Find 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]
Let ABCDEFHIJKLM be a regular dodecagon with center G and circumradius 11. Line O is the angle bisector of angle FGM. Line S passes through point L perpendicular to line O. Point Z is the reflection of point B across line S. Let T be the midpoint of segment CZ. Find the length of segment TK. 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 10. Let P be the perpendicular bisector of segment BA. Line Z passes through point G parallel to line P. Point S is the reflection of point D across line Z. What is the length of segment LS? 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 4. Line V passes through point A parallel to segment BH. Point L is the reflection of point H across line V. Point Q is the midpoint of segment AG. What is the distance between points Q and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDFGHI is a regular octagon with center E and circumradius 6. Let R be the circumcircle of polygon ABCDFGHI. Let M be the reflection of circle R through point I. Let N be the reflection of point H across point A. Point W is the inversion of point N with respect to circle M. 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 ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 4. Line W is the perpendicular bisector of segment MB. Line O passes through point C perpendicular to line W. Let Y be the circumcircle of polygon ABCDEFGIJKLM. Let N be the reflection of circle Y across line O. Circle S is the reflection of circle N across point G. Point P is the midpoint of segment GK. Point T is the inversion of point P with respect to circle S. Find the length of segment TD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HS = 6. Let ABCDEFGI be a regular octagon with center S and circumradius 1. Let K be the reflection of point D across point E. Let W be the circumcircle of polygon ABCDEFGI. Let Z be the inversion of point K in circle W. Point P is the reflection of point F across segment ZA. Find the distance between points E and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center W and circumradius 1. Let M be the midpoint of segment EI. Line Y is the perpendicular bisector of segment IJ. Let V be the line through point I parallel to line Y. Point X is the reflection of point E across line V. What is the measure of angle MIX, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center W and circumradius 6. Let H be the perpendicular bisector of segment ED. Let T be the reflection of line H through point C. Point P is the reflection of point G across line T. Point J is the midpoint of segment PB. Find the length of segment JD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center T and circumradius 5. Let W be the line through point F parallel to segment BT. Line S passes through point G parallel to segment FJ. Lines S and W intersect at point Q. Find the measure of angle IQE, 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 UI = 6. Let ABCDE be a regular 5-gon with center I and circumradius 4. Let N be the line through points I and D. Let T be the line through point B perpendicular to line N. Let R be the line through point E perpendicular to line T. Point M is the reflection of point B across line R. Let Z be the circumcircle of polygon ABCDE. Point F lies on line T. Let L be the inversion of point M in circle Z. Find the measure of angle FML, 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 ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 2. Point R is the reflection of point K across point C. Let Q be the angle bisector of angle IFM. Point Z is obtained by rotating point B by angle 2pi/3 about point K. Let X be the reflection of line Q through point J. Line Y is the reflection of line X across line Q. Line T is the reflection of line Y across point R. Let S be the reflection of line T across line Y. Point P is the reflection of point Z across line S. What is the length of segment HP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDFGHI be a regular octagon with center E and circumradius 2. Let J be the line through points E and A. Let Y be the reflection of line J through point I. Point Z is the midpoint of segment GC. Line K passes through point I perpendicular to segment GC. Line P is the reflection of line K across line Y. Point O is the reflection of point B across line P. 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 ACDEFG be a regular hexagon with center B and circumradius 11. Point P is obtained by rotating point E by angle 5pi/12 about point G. Let I be the circumcircle of polygon ACDEFG. Let S be the inversion of point P in circle I. Circle O is the reflection of circle I across point E. Point V is the midpoint of segment CG. Let N be the inversion of point V in circle O. Find the distance between points N and S. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center S and circumradius 3. Let U be the rotation of point S by angle pi/6 around point H. Point Y is the midpoint of segment BU. Let V be the midpoint of segment US. Let Z be the midpoint of segment AY. Find the length of segment ZV. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center X and circumradius 5. Point L is the reflection of point F across point D. Let W be the midpoint of segment LC. Line T is the angle bisector of angle GFL. Point Y is the reflection of point X across line T. Let Q be the midpoint of segment YW. Find the distance between points H and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center Q and circumradius 7. Let U be the midpoint of segment QG. Let T be the midpoint of segment BQ. Let P be the reflection of point J across point D. Point N is the reflection of point P across segment UT. What is the distance between points N and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 7. Let X be the reflection of point I across point J. Line P is the perpendicular bisector of segment FD. Line Q is the reflection of line P across point J. Let Y be the reflection of line Q through point X. Point U is the reflection of point L across line Y. What is the distance between points L and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DA = 7. Let BCEFGHIJKLMN be a regular dodecagon with center A and circumradius 8. Point U is the reflection of point N across segment EJ. Point V is the reflection of point U across segment JI. What is the length of segment VN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDFG is a regular hexagon with center E and circumradius 12. Let L be the midpoint of segment CD. Let O be the reflection of point D across point A. Let V be the perpendicular bisector of segment LA. Line M passes through point O perpendicular to line V. Point I is the reflection of point G across line M. Find the distance between points B 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 W and circumradius 10. Let T be the perpendicular bisector of segment EK. Line P passes through point W perpendicular to line T. Line O is the reflection of line P across point F. Point Z is the reflection of point W across line O. What is the length of segment ZJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFG be a regular 7-gon with center T and circumradius 12. Let X be the midpoint of segment CF. Let W be the rotation of point F by angle 3pi/4 around point X. Line I is the angle bisector of angle TGA. Let U be the reflection of point C across point W. Point J is the reflection of point U across line I. Line V passes through point B parallel to segment DA. Point L is the reflection of point F across line V. Find the length of segment LJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SZ = 5. ABCDEFGH is a regular octagon with center S and circumradius 10. Point V is the reflection of point Z across point A. Point T is the reflection of point S across point V. Let K be the perpendicular bisector of segment GT. Point I is the reflection of point A across line K. What is the distance between points T and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center W and circumradius 4. Let I be the perpendicular bisector of segment CW. Point Q is the reflection of point G across line I. Point U is the reflection of point W across point E. Point Z is the reflection of point U across line I. Let V be the reflection of point B across point Z. What is the distance between points V and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center J and circumradius 1. Line U is the angle bisector of angle JBE. Line M passes through points A and B. Line X is the reflection of line U across line M. Point Z is the reflection of point D across line X. What is the distance between points D 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 ABCDE be a regular 5-gon with center G and circumradius 4. Point X is obtained by rotating point E by angle pi/3 about point G. Line V is the angle bisector of angle GXA. Let O be the circumcircle of polygon ABCDE. Circle N is the reflection of circle O across line V. Let M be the inversion of point G in circle N. Find the distance between points M and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHJKL is a regular 11-gon with center I and circumradius 8. Line R is the perpendicular bisector of segment AL. Let N be the rotation of point D by angle pi/2 around point G. Line X passes through points H and N. Let U be the reflection of line R across line X. Point T is the reflection of point K across segment FB. Line P passes through point T perpendicular to line U. Point Q is the reflection of point J across line P. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDE be a square with center B and circumradius 9. Let Z be the reflection of point C across point D. Point T is the reflection of point C across point E. Point Q is the midpoint of segment ZE. Find the measure of angle TZQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 11. Line Z is the angle bisector of angle AGI. Point O is the reflection of point I across line Z. Point T is the reflection of point O across segment JM. 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]
ABCDEFGHIJK is a regular 11-gon with center O and circumradius 1. Point S is the midpoint of segment DI. Point M is the reflection of point H across point S. Point T is the midpoint of segment KH. Find the distance between points T 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 ABCDEF be a regular hexagon with center V and circumradius 3. Let Z be the circumcircle of polygon ABCDEF. Line H is the angle bisector of angle AFD. Circle G is the reflection of circle Z across line H. Point M is the inversion of point E with respect to circle G. What is the length of segment MD? 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 9. Let J be the midpoint of segment CA. Let T be the reflection of point D across point A. Point R is the midpoint of segment JF. What is the distance between points T 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 M and circumradius 2. Let R be the angle bisector of angle IAM. Line Z passes through point I perpendicular to line R. Point W is the reflection of point G across line Z. Let P be the midpoint of segment IW. What is the distance between points C and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center T and circumradius 3. Let J be the rotation of point B by angle pi/12 around point D. Line Q passes through points E and B. Let W be the line through point F parallel to line Q. Let N be the reflection of line W through point B. Line Y passes through points J and T. Point R is the intersection of lines Y and N. Find the length of segment RF. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center S and circumradius 1. Let U be the circumcircle of polygon ABCDEF. Let I be the reflection of circle U through point E. Let Y be the reflection of circle I through point A. Point K is the inversion of point S with respect to circle Y. Line J is the angle bisector of angle KSC. Point X is the reflection of point K across line J. What is 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 ABCDEF be a regular hexagon with center I and circumradius 11. Let Q be the circumcircle of polygon ABCDEF. Line M is the perpendicular bisector of segment FC. Circle J is the reflection of circle Q across point A. Circle N is the reflection of circle J across line M. Let W be the inversion of point I in circle N. Find the measure of angle CIW, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center Y and circumradius 5. Let V be the reflection of point Y across point C. Let S be the perpendicular bisector of segment CA. Line K is the reflection of line S across point V. Point O is the reflection of point A across line K. Find the distance between points D 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 JU = 11. Let ABCDEF be a regular hexagon with center U and circumradius 5. Point P is the midpoint of segment DC. Let G be the line through points U and D. Let O be the line through point P parallel to line G. Let I be the line through points D and B. Lines O and I intersect at point S. What is the distance between points E and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center U and circumradius 4. Line S is the perpendicular bisector of segment AF. Let O be the reflection of point A across point B. Point X is the reflection of point O across line S. Let I be the perpendicular bisector of segment XF. Point Q is the reflection of point A across line I. Find the distance between points E 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 ABCEF be a regular 5-gon with center D and circumradius 11. Let H be the midpoint of segment BE. Point G is the midpoint of segment CE. Let R be the midpoint of segment CH. What is 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]
Let ABCDEF be a regular hexagon with center G and circumradius 7. Line S is the perpendicular bisector of segment DA. Let Q be the reflection of line S through point F. Let Z be the reflection of point G across point D. Point X is the reflection of point Z across line Q. What is the distance between points X 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 Q and circumradius 1. Point U is the reflection of point G across point A. Line V passes through points U and A. Let R be the reflection of point K across point G. Line X passes through point R parallel to line V. Line W passes through points K and A. Line O is the perpendicular bisector of segment RD. Let M be the reflection of line O across line W. Lines X and M intersect at point Z. Find the distance between points Z and B. 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 S and circumradius 10. Point P is obtained by rotating point S by angle pi/2 about point G. Point T is the reflection of point A across point S. Point L is the reflection of point T across point I. Point R is the reflection of point P across segment DI. Line Y passes through point R perpendicular to segment DI. Point U is the reflection of point L across line Y. Point Z is the reflection of point U across segment DI. What is the distance between points Z and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center X and circumradius 12. Point M is the reflection of point C across point H. Let T be the reflection of point E across point A. Point R is the reflection of point T across segment EM. Find the length of segment FR. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
CR = 7. ABDEF is a regular 5-gon with center R and circumradius 2. Point U is the midpoint of segment FD. Point P is the reflection of point B across segment ED. Let Z be the midpoint of segment UF. What is the distance between points P and Z? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ABCDEF be a regular hexagon with center N and circumradius 12. Let Q be the perpendicular bisector of segment EB. Let T be the reflection of point N across point D. Line R is the perpendicular bisector of segment FT. Point S is on line Q. Line L passes through point S parallel to line R. Line K passes through point C parallel to segment AN. Lines K and L intersect at point P. What is the measure of angle PBN, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEF is a regular 5-gon with center D and circumradius 4. Line V is the angle bisector of angle ADF. Point T is the reflection of point C across point F. Let J be the line through point T perpendicular to line V. Let R be the line through point E parallel to line J. Let K be the perpendicular bisector of segment BT. Point Z is the reflection of point C across line R. Line L is the reflection of line V across line K. Point U is the reflection of point B across line L. 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]
Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 12. Line S is the angle bisector of angle LIA. Let U be the perpendicular bisector of segment KE. Let W be the reflection of line U across line S. Point P is the reflection of point J across line S. Point T is the reflection of point P across line W. Let O be the midpoint of segment LB. What is the distance between points O and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center F and circumradius 1. Let Z be the line through points D and A. Line V passes through point E parallel to line Z. Point Y is the reflection of point B across point C. Point N is the reflection of point F across line V. Let S be the reflection of point Y across point N. Find the distance between points Y 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 BCDEFGHI be a regular octagon with center A and circumradius 10. Point O is the midpoint of segment AI. Point K is the midpoint of segment BO. Point L is the reflection of point A across segment IB. What is the distance between points K and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center M and circumradius 5. Point V is obtained by rotating point E by angle 2pi/3 about point A. Line U is the angle bisector of angle EFD. Point J is the reflection of point M across line U. Point H is the reflection of point J across point C. Line N is the perpendicular bisector of segment HD. Let T be the circumcircle of polygon ABCDEF. Point S is the reflection of point V across line N. Point X is the inversion of point S with respect to circle T. What is the distance between points M 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 I and circumradius 11. Let L be the reflection of point C across point A. Point Z is the midpoint of segment AG. Let P be the line through point G perpendicular to segment LD. Point O is the reflection of point Z across line P. Find the distance between points D 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 Q and circumradius 9. Let K be the midpoint of segment BC. Point Y is the midpoint of segment KD. Let P be the midpoint of segment CQ. 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 ABCEFG be a regular hexagon with center D and circumradius 8. Point W is the midpoint of segment GF. Let J be the perpendicular bisector of segment FC. Point X is the reflection of point W across line J. Point O is the midpoint of segment AX. What is the distance between points C 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 ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 4. Point Z is the reflection of point E across segment IL. Let P be the midpoint of segment HM. What is the length of segment ZP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center K and circumradius 1. Let X be the midpoint of segment CA. Let J be the line through point D perpendicular to segment EC. Let Y be the line through point X perpendicular to line J. Point N is the reflection of point A across line Y. Find the distance between points N and E. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center P and circumradius 9. Point W is obtained by rotating point P by angle pi/3 about point F. Point L is the reflection of point G across segment DW. Point R is the midpoint of segment DW. Point T is the midpoint of segment LR. Find the distance between points T and D. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEFGHIJ is a regular 10-gon with center U and circumradius 10. Let P be the line through points I and U. Line N passes through point J parallel to line P. Point R is the intersection of segment AU and line N. Find the measure of angle RIC, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ABCDEF be a regular hexagon with center I and circumradius 3. Point L is the reflection of point E across point C. Let Z be the line through points A and F. Let V be the circumcircle of polygon ABCDEF. Let N be the reflection of circle V through point A. Circle Y is the reflection of circle N across line Z. Point M is the inversion of point L with respect to circle Y. What is 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]
ABCDE is a regular 5-gon with center J and circumradius 2. Line G passes through points J and C. Line Q passes through points A and E. Point K is the intersection of lines G and Q. Let R be the midpoint of segment KB. What is the length of segment RK? 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 O and circumradius 6. Let Y be the circumcircle of polygon ABCDEFGHIJK. Point V is the reflection of point O across point A. Line S is the perpendicular bisector of segment OV. Circle U is the reflection of circle Y across line S. Let L be the inversion of point C in circle U. Find the distance between points L 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 ABCE be a square with center D and circumradius 9. Point U is the reflection of point C across point E. Let Q be the midpoint of segment UD. Point J is the reflection of point A across segment UQ. Find the length of segment UJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEFG be a regular hexagon with center D and circumradius 7. Let J be the reflection of point C across point G. Let Y be the midpoint of segment FE. Let H be the circumcircle of polygon ABCEFG. Let U be the reflection of circle H through point Y. Let V be the inversion of point B in circle U. What is 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]
Let ABCDE be a regular 5-gon with center S and circumradius 1. Let Y be the line through points E and A. Point Z is the midpoint of segment SD. Line P is the perpendicular bisector of segment BD. Let G be the reflection of line P through point Z. Point I is the intersection of lines Y and G. What is the distance between points I and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center S and circumradius 2. Let V be the rotation of point L by angle pi/12 around point J. Let P be the line through points V and C. Let U be the line through point S parallel to line P. Let R be the reflection of line U through point A. Point W is the reflection of point C across line R. Find the length of segment WD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHI is a regular 9-gon with center L and circumradius 2. Line X passes through points B and A. Line R is the reflection of line X across point E. Point Z is the reflection of point H across line R. Point J is the reflection of point F across point Z. What is the distance between points G and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center Z and circumradius 11. Line L is the angle bisector of angle FAC. Line T is the reflection of line L across point F. Point P is the reflection of point B across line T. Point W is the midpoint of segment PE. Point J is the midpoint of segment AW. Find the length of segment AJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEF is a regular 5-gon with center A and circumradius 4. Point V is the reflection of point B across point E. Let I be the midpoint of segment BF. Line M is the perpendicular bisector of segment IA. Point O is the reflection of point B across line M. Point Q is the reflection of point V across point O. Point K is the midpoint of segment BQ. Find the length of segment OK. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFG be a regular hexagon with center C and circumradius 10. Let W be the rotation of point B by angle 5pi/12 around point D. Point J is the reflection of point A across point F. Line H passes through points E and W. Point N is the reflection of point J across line H. Point Z is the midpoint of segment WA. Find the length of segment ZN. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center L and circumradius 3. Let W be the midpoint of segment AB. Point H is the reflection of point A across segment WE. Find the distance between points H 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 ABCDEFGHI be a regular 9-gon with center Z and circumradius 10. Line X passes through point B perpendicular to segment AD. Point J is the reflection of point F across line X. Point O is the midpoint of segment AD. Find the length of segment JO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFG is a regular hexagon with center C and circumradius 4. Let I be the midpoint of segment FD. Let X be the midpoint of segment BI. Line T passes through points A and X. Point V is the intersection of line T and segment FD. Let H be the circumcircle of polygon ABDEFG. Let Q be the inversion of point V in circle H. Find the length of segment DQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFGHIJK be a regular 10-gon with center B and circumradius 3. Point X is the reflection of point K across segment FG. Let O be the midpoint of segment FG. Let Z be the angle bisector of angle CXD. Let S be the circumcircle of polygon ACDEFGHIJK. Let W be the reflection of circle S across line Z. Let Y be the inversion of point D in circle W. Find the distance between points O and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center Z and circumradius 2. Line O passes through points E and L. Let R be the line through point C parallel to segment GK. Point P is the intersection of lines R and O. Point W is the reflection of point P across segment GK. What is the length of segment WE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center K and circumradius 6. Let T be the line through point C perpendicular to segment BC. Let I be the reflection of point C across point D. Point W is the reflection of point K across line T. What is the distance between points W and I? 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 V and circumradius 3. Let N be the midpoint of segment DB. Let W be the line through points A and D. Point K is the reflection of point N across line W. Let M be the reflection of point V across point F. 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 ABCDEF be a regular hexagon with center P and circumradius 12. Let V be the midpoint of segment ED. Let J be the circumcircle of polygon ABCDEF. Point N is the inversion of point V with respect to circle J. Point G is the reflection of point C across segment NB. What is the length of segment NG? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ACDEFGHI is a regular octagon with center B and circumradius 6. Point Z is the reflection of point F across point H. Let T be the line through points H and E. Let Q be the line through point D parallel to line T. Let N be the reflection of line Q across line T. Let J be the reflection of point E across point I. Point R is the reflection of point J across line N. What is the distance between points R 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 ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 1. Point Z is the midpoint of segment LK. Let Q be the circumcircle of polygon ABCDEFGHIJKL. Point R is the reflection of point B across point Z. Let U be the inversion of point R in circle Q. What is the distance between points L and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center U and circumradius 1. Point V is the reflection of point K across segment JC. Let P be the reflection of point V across point J. Point N is the reflection of point P across segment JC. What is the length of segment NC? 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 6. Let F be the perpendicular bisector of segment DU. Let R be the reflection of line F through point D. Let G be the reflection of line F across line R. Point H is the reflection of point E across line G. Point Q is the midpoint of segment DU. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 10. Line O is the perpendicular bisector of segment EJ. Let V be the reflection of line O through point L. Let Y be the reflection of line O across line V. Point P is the reflection of point D across line Y. Let S be the line through points H and P. Lines S and Y intersect at point U. Find the distance between points G and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center S and circumradius 7. Let K be the midpoint of segment DS. Let O be the midpoint of segment DC. 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]
ACDEFGHI is a regular octagon with center B and circumradius 2. Let O be the reflection of point C across point G. Let P be the reflection of point G across point E. Point J is the reflection of point P across segment OD. Find the distance between points J 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 L and circumradius 2. Let K be the midpoint of segment BA. Line O is the angle bisector of angle FBK. Line X passes through point A perpendicular to line O. Point J is the reflection of point F across line X. Find the distance between points E 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 ABDEF be a regular 5-gon with center C and circumradius 2. Point X is obtained by rotating point B by angle pi/3 about point A. Let G be the circumcircle of polygon ABDEF. Let T be the inversion of point X in circle G. Point Y is the reflection of point B across point T. Let K be the midpoint of segment AC. Find 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]
ABCDEFGHIJ is a regular 10-gon with center X and circumradius 11. Let S be the midpoint of segment JX. Let W be the rotation of point E by angle 2pi/3 around point S. Point L is obtained by rotating point I by angle 5pi/6 about point C. Let O be the angle bisector of angle WXB. Let Y be the reflection of line O through point L. Let M be the reflection of line O across line Y. Point N is the reflection of point W across line M. Find 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 ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 4. Line S passes through points F and I. Let X be the line through points M and F. Line O is the reflection of line S across line X. Line R is the reflection of line O across point C. Point U is the intersection of line R and segment LA. What is the length of segment CU? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ACDE be a square with center B and circumradius 1. Let O be the midpoint of segment BC. Point M is the reflection of point D across point E. Point I is the reflection of point M across point O. Point T is the reflection of point E across segment MA. What is the distance between points T and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFGHI be a regular octagon with center C and circumradius 12. Point S is the midpoint of segment HC. Let M be the reflection of point F across point S. Point U is the reflection of point M across segment HG. Find the length of segment IU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center K and circumradius 8. Let J be the circumcircle of polygon ABCDEF. Line N passes through point C parallel to segment DA. Let W be the line through point D perpendicular to line N. Circle L is the reflection of circle J across line W. Let X be the inversion of point E in circle L. Find the distance between points F 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 ABCDEFGHIJK be a regular 11-gon with center N and circumradius 12. Let L be the angle bisector of angle JIN. Let M be the angle bisector of angle GDF. Line V is the reflection of line M across line L. Let Q be the line through points C and I. Lines V and Q intersect at point R. Find the distance between points N and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center M and circumradius 11. Let Y be the line through points B and M. Point N lies on line Y at distance 2 from point B. Let Q be the midpoint of segment MN. Line W passes through points Q and A. Let L be the reflection of point M across point D. Point S lies on line W. Let X be the midpoint of segment BS. Point R is the reflection of point L across segment EX. Find the length of segment ER. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center O and circumradius 5. Line Z is the perpendicular bisector of segment EB. Let Q be the reflection of line Z through point E. Let U be the line through point B parallel to line Q. Point N is the reflection of point C across line U. What is the distance between points N and F? 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 T and circumradius 10. Let V be the reflection of point D across point B. Let Q be the circumcircle of polygon ABCDE. Let S be the reflection of circle Q through point A. Point I is the inversion of point E with respect to circle S. What is the length of segment IV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEF is a regular 5-gon with center C and circumradius 11. Point T is the midpoint of segment DC. Let S be the angle bisector of angle DEA. Point P is the reflection of point T across line S. Let X be the reflection of point E across point B. What is the length of segment PX? 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 P and circumradius 10. Let Q be the circumcircle of polygon ABCDEFGHIJKL. Point W is the midpoint of segment CF. Circle V is the reflection of circle Q across point B. Point O is the inversion of point L with respect to circle V. What is the length of segment OW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center O and circumradius 1. Let G be the midpoint of segment AB. Let Z be the midpoint of segment AD. What is the measure of angle AGZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ABCDEF be a regular hexagon with center K and circumradius 10. Point V is the reflection of point F across point D. Point L is the reflection of point V across segment KC. Let Q be the perpendicular bisector of segment KC. Point X is the reflection of point L across line Q. Let U be the midpoint of segment KC. Find the distance between points X and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center T and circumradius 8. Let U be the rotation of point A by angle pi/12 around point C. Let K be the midpoint of segment DU. Line E passes through points C and K. Line R is the reflection of line E across point A. Point S is the reflection of point K across line R. Find the length of segment SD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFGHIJ is a regular 9-gon with center D and circumradius 6. Line K is the angle bisector of angle EAI. Line T passes through point H perpendicular to line K. Point M is the reflection of point H across point D. Point O is the reflection of point J across point M. Point P is the reflection of point H across point A. Point R is the reflection of point P across line T. Find the distance between points R and O. Express your answer as a decimal number rounded to 2 decimal places.

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