[User]
Let ABCDEFGHIJ be a regular 10-gon with center X and circumradius 9. Point P is obtained by rotating point C by angle pi/4 about point D. Let M be the angle bisector of angle FPH. Line U passes through point E parallel to line M. Point Q is the reflection of point A across line U. Point V is the midpoint of segment PA. 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}

dataset: MATH


[User]
ABCDEF is a regular hexagon with center L and circumradius 6. Let Z be the line through points B and F. Let O be the midpoint of segment LC. Let Y be the line through point O parallel to line Z. Segment BE and line Y intersect at point S. Find the distance between points S and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center N and circumradius 12. Line X is the perpendicular bisector of segment NA. Line T is the reflection of line X across point A. Let Z be the reflection of line T across line X. Point W is the reflection of point N across point C. Let J be the reflection of line Z across line T. Point R is the reflection of point W across line J. Find the distance between points W and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDFGHIJK is a regular 10-gon with center E and circumradius 3. Let Z be the reflection of point B across point C. Let U be the midpoint of segment IZ. Line O passes through point B parallel to segment HK. Point M is the reflection of point U across segment AZ. Point R is the reflection of point M across line O. Find the distance between points U and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center Q and circumradius 4. Point J is the midpoint of segment CQ. Point X is the midpoint of segment DE. Line M passes through point J perpendicular to segment DE. Point U is the reflection of point X across line M. Find the length of segment AU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHI be a regular 9-gon with center N and circumradius 11. Line V passes through points H and B. Let L be the line through points E and D. Let W be the reflection of point E across point F. Point U is the intersection of lines V and L. Point X is the reflection of point C across point U. Let O be the midpoint of segment WC. Find the distance between points X and O. 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 9. Let Q be the midpoint of segment AD. Line T passes through points C and Q. Point S is the reflection of point D across line T. Point W is the reflection of point Q across point S. What is the distance between points W and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJK is a regular 11-gon with center N and circumradius 11. Let M be the reflection of point D across point G. Let V be the angle bisector of angle DFE. Let R be the line through point N perpendicular to line V. Point X is the reflection of point F across point H. Point L is the reflection of point M across line R. What is the distance between points L 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 ABCDEFGI be a regular octagon with center H and circumradius 4. Line L is the perpendicular bisector of segment HF. Line O passes through point D perpendicular to line L. Point P is the reflection of point B across line L. Point Y is the reflection of point A across line O. Find 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 ABCDEF be a regular hexagon with center Z and circumradius 4. Line N passes through points F and A. Point M is the reflection of point B across line N. Point Q is the midpoint of segment ED. Point X is the midpoint of segment FM. Find the distance between points X and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center T and circumradius 7. Let G be the reflection of point T across point B. Let O be the midpoint of segment DG. Point Z is the midpoint of segment TA. What is the length of segment OZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDFGH is a regular 7-gon with center E and circumradius 2. Line T is the perpendicular bisector of segment HE. Line J passes through points H and C. Let Z be the reflection of line T through point G. Lines J and Z intersect at point Q. Find the distance between points F 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 ACDEFG be a regular hexagon with center B and circumradius 2. Point U is obtained by rotating point E by angle pi/12 about point D. Let Y be the circumcircle of polygon ACDEFG. Circle L is the reflection of circle Y across point U. Let P be the reflection of circle L through point C. Point S is the inversion of point F with respect to circle P. 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 ABCE be a square with center D and circumradius 7. Line T is the angle bisector of angle BEC. Let N be the circumcircle of polygon ABCE. Point K is the reflection of point C across point B. Circle S is the reflection of circle N across line T. Let U be the inversion of point C in circle S. Find the distance between points K 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 ABCDEFGHIJLM be a regular dodecagon with center K and circumradius 4. Let R be the rotation of point F by angle 5pi/12 around point B. Line N passes through point R perpendicular to segment FC. Point V is the reflection of point D across line N. Let Q be the reflection of line N through point V. Point O is the reflection of point J across line Q. Find 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]
ABCD is a square with center I and circumradius 2. Let Q be the rotation of point D by angle 5pi/12 around point B. Point W is the midpoint of segment QI. Let N be the reflection of point A across point D. Point K is the reflection of point W across segment QN. What is the distance between points B 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 Q and circumradius 1. Let U be the circumcircle of polygon ABCDEFGH. Let O be the reflection of point C across point D. Circle K is the reflection of circle U across point E. Let V be the inversion of point O in circle K. Find the distance between points V and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center V and circumradius 2. Line M passes through points B and E. Line W passes through points E and V. Let O be the reflection of line M across line W. Point Y is the reflection of point G across point E. Let J be the reflection of line W through point Y. Let R be the midpoint of segment FD. Point K is the intersection of lines J and O. What is the distance between points K and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center M and circumradius 5. Let Z be the line through points D and G. Let U be the line through points D and E. Line R is the reflection of line Z across line U. Point N is the reflection of point G across line R. What is the measure of angle NAM, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center P and circumradius 12. Let J be the circumcircle of polygon ABCDEFGH. Let U be the angle bisector of angle PCE. Point T is the midpoint of segment BP. Point V is the reflection of point E across line U. Line K is the reflection of line U across point T. Let O be the reflection of circle J across line K. Point X is the inversion of point H with respect to circle O. Point S is the reflection of point V across segment AB. What is the distance between points X and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center H and circumradius 2. Let U be the reflection of point C across point B. Let N be the line through points H and C. Line J passes through points D and A. Lines J and N intersect at point Q. Find the measure of angle QUC, 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 P and circumradius 3. Let W be the reflection of point F across point C. Point I is the reflection of point A across segment EW. Point U is the reflection of point C across segment EW. Find the length of segment UI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFG be a regular 7-gon with center P and circumradius 10. Point Q is the reflection of point C across segment FB. Let W be the line through point F perpendicular to segment FB. Point I is the reflection of point Q across line W. Find the length of segment EI. 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 9. Point T is the reflection of point I across segment GJ. Let Y be the line through point T parallel to segment GJ. Point L is the reflection of point C across line Y. Find the length of segment LE. 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 3. Line L is the angle bisector of angle BKA. Point J lies on line L at distance 8 from point D. Let Z be the perpendicular bisector of segment DA. Point S is the reflection of point E across segment JB. Point I is the reflection of point S across line Z. What is the length of segment CI? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center T and circumradius 6. Let Q be the reflection of point E across point B. Let O be the reflection of point B across point D. Let U be the circumcircle of polygon ABCDEFGH. Line P is the angle bisector of angle HOQ. Let V be the reflection of circle U across line P. Let W be the reflection of circle V through point C. Let S be the inversion of point C in circle W. What is the length of segment SA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center X and circumradius 8. Let S be the circumcircle of polygon ABCDEFGH. Let Z be the midpoint of segment BX. Let J be the inversion of point Z in circle S. What is the length of segment JH? 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 5. Let Q be the midpoint of segment CF. Let O be the midpoint of segment DE. Line H passes through points Q and F. Point X is the reflection of point D across line H. What is the distance between points X 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 ABCDEF be a regular hexagon with center Q and circumradius 1. Line O passes through points B and E. Let R be the midpoint of segment CE. Point S is the reflection of point R across line O. Point M is the reflection of point S across point C. Find the distance between points M and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDFGH is a regular 7-gon with center E and circumradius 7. Let W be the reflection of point C across point B. Point X is the midpoint of segment GE. Let M be the angle bisector of angle WCX. Point P is the reflection of point A across line M. Let L be the perpendicular bisector of segment PH. Point Y is the reflection of point A across line L. What is the length of segment CY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEG is a regular hexagon with center F and circumradius 2. Let N be the circumcircle of polygon ABCDEG. Let M be the midpoint of segment FG. Let S be the reflection of circle N through point M. Point O is the inversion of point B with respect to circle S. Let J be the inversion of point O in circle N. Find the distance between points J and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center X and circumradius 5. Line M is the perpendicular bisector of segment XC. Point R is the reflection of point A across line M. Point F is the reflection of point B across point D. Point S is the midpoint of segment RC. Find the length of segment SF. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center V and circumradius 6. Let Y be the midpoint of segment FA. Let I be the circumcircle of polygon ABCDEF. Point S is the inversion of point Y with respect to circle I. Let N be the midpoint of segment SE. What is the distance between points N 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 P and circumradius 12. Let L be the rotation of point F by angle pi/12 around point I. Line X is the perpendicular bisector of segment GF. Let T be the line through point L parallel to line X. Point V is the intersection of segment PD and line T. Find the length of segment EV. 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 10. Point S is the reflection of point E across point B. Line X passes through points S and F. Let O be the midpoint of segment CD. Point U is the reflection of point O across line X. What is the measure of angle UFO, 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 Y and circumradius 11. Line Q is the perpendicular bisector of segment AG. Let X be the reflection of line Q through point C. Let L be the line through point E parallel to line X. Point R is the reflection of point H across line L. Find the length of segment RB. 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 11. Let H be the rotation of point B by angle pi/12 around point A. Let U be the perpendicular bisector of segment XH. Point V is the reflection of point D across line U. Let Q be the midpoint of segment VX. What is the distance between points Q 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 ABCDFGHIJK be a regular 10-gon with center E and circumradius 3. Line O passes through points J and C. Point T is the intersection of line O and segment AE. Point X is the midpoint of segment DT. Find the length of segment KX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center X and circumradius 4. Let R be the rotation of point A by angle pi/4 around point E. Line V is the perpendicular bisector of segment RF. Let Q be the reflection of line V through point C. Point M is the reflection of point E across line Q. Find the length of segment FM. 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 2. Let L be the circumcircle of polygon ABCDEF. Let O be the reflection of point F across point B. Point Z is the midpoint of segment OA. Point T is the inversion of point Z with respect to circle L. What is the length of segment TC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center U and circumradius 3. Let V be the midpoint of segment DA. Point H is the midpoint of segment CB. Line K is the perpendicular bisector of segment DV. Point S is the reflection of point U across line K. Find the measure of angle VHS, 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 T and circumradius 2. Let S be the angle bisector of angle TCB. Let N be the midpoint of segment DA. Let E be the line through points T and D. Let K be the reflection of line E across line S. Point M is the reflection of point A across line K. Find the length of segment NM. 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 4. Let Q be the angle bisector of angle EIF. Let P be the midpoint of segment JH. Point Z is the reflection of point P across line Q. Line V passes through point L parallel to line Q. Point X is the reflection of point B across line V. Let O be the perpendicular bisector of segment XZ. Point U is the reflection of point C across line O. Find the length of segment UL. 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 7. Line T is the angle bisector of angle EYA. Point X is on line T with |XY| = 5. Let I be the reflection of line T through point E. Let G be the line through point E parallel to line I. Line F is the reflection of line G across point X. Point M is the reflection of point E across line F. Find the distance between points A 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 X and circumradius 1. Let M be the perpendicular bisector of segment CD. Let Q be the reflection of line M through point E. Point W is the reflection of point A across line Q. What is the measure of angle WFG, in radians? 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 K be the circumcircle of polygon ACDEFGHI. Point N is the reflection of point F across point G. Circle O is the reflection of circle K across point A. Let V be the reflection of circle O through point D. Let P be the inversion of point N in circle V. What is the length of segment PC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center V and circumradius 8. Point W is the reflection of point V across point D. Point O is the midpoint of segment VC. Point Q is the reflection of point H across segment WC. 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 ABCDEF be a regular hexagon with center T and circumradius 12. Let X be the midpoint of segment FD. Point Z is the reflection of point B across point D. Line N passes through point Z perpendicular to segment DF. Point H is the reflection of point A across line N. Find the length of segment HX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFG be a regular 7-gon with center K and circumradius 11. Point R is obtained by rotating point F by angle pi/3 about point B. Point Y is the midpoint of segment KC. Point P is the reflection of point A across point C. Let X be the midpoint of segment YR. Let S be the reflection of point X across point C. What is the length of segment PS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center N and circumradius 11. Let T be the perpendicular bisector of segment AE. Let I be the reflection of line T through point E. Line L is the reflection of line T across line I. Line M is the perpendicular bisector of segment NH. Let U be the reflection of line M across line L. Point O is the reflection of point D across line U. Find the distance between points E 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 ABCE be a square with center D and circumradius 4. Line V passes through points E and A. Point H is the reflection of point C across line V. Point Y is the reflection of point B across line V. Point R is the midpoint of segment HY. Find the distance between points R 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 G and circumradius 3. Let T be the circumcircle of polygon ABCDEF. Let W be the midpoint of segment CE. Point U is the midpoint of segment BD. Point Q is the inversion of point U with respect to circle T. Let L be the midpoint of segment WA. What is the length of segment QL? 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 U and circumradius 9. Let L be the circumcircle of polygon ABCDEFGHIJ. Circle M is the reflection of circle L across point H. Circle T is the reflection of circle M across point C. Point Q is the inversion of point I with respect to circle T. What is 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]
ABDEFGH is a regular 7-gon with center C and circumradius 4. Let V be the midpoint of segment EF. Let J be the line through point E perpendicular to segment VC. Point Q is the reflection of point G across line J. What is the measure of angle QEA, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center J and circumradius 6. Let T be the line through point E parallel to segment DB. Let L be the line through point H parallel to line T. Let P be the circumcircle of polygon ABCDEFGH. Circle U is the reflection of circle P across line L. Point K is the inversion of point E with respect to circle U. Find the distance between points D and K. 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 9. Line N is the perpendicular bisector of segment HF. Let Q be the line through point E parallel to line N. Point J is the reflection of point G across line Q. What is the distance between points J and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center K and circumradius 7. Point I is the reflection of point C across point D. Let Z be the reflection of point C across point I. Let H be the circumcircle of polygon ABCD. Let W be the inversion of point Z in circle H. Find 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]
Let ABCDEF be a regular hexagon with center X and circumradius 12. Line P passes through points A and B. Point R is the reflection of point F across line P. Let N be the midpoint of segment XF. What is the length of segment RN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 2. Let N be the midpoint of segment IB. Let X be the circumcircle of polygon ABCDEFGHIJKL. Circle V is the reflection of circle X across point N. Point P is the inversion of point G with respect to circle V. What is the measure of angle IPS, 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 ABCDFG be a regular hexagon with center E and circumradius 10. Let U be the midpoint of segment BD. Point R is the midpoint of segment CU. Line W passes through points G and R. Segment AF and line W intersect at point S. What is the distance between points S and G? 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 11. Let M be the reflection of point F across point I. Let S be the reflection of point E across point B. Point Z is the midpoint of segment JS. Point U is the reflection of point Z across point C. Find the measure of angle UBM, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFGH is a regular 7-gon with center C and circumradius 8. Let Y be the angle bisector of angle AGB. Let J be the reflection of line Y through point C. Let N be the midpoint of segment AE. Point Q is the reflection of point N across line J. Point X is the midpoint of segment CQ. 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]
Let ABCDEFGHI be a regular 9-gon with center R and circumradius 6. Point M is the reflection of point C across point E. Let J be the circumcircle of polygon ABCDEFGHI. Point U is the inversion of point M with respect to circle J. Let P be the midpoint of segment GH. What is the distance between points P 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 ABCDEFGH be a regular octagon with center L and circumradius 7. Point N is obtained by rotating point E by angle pi/2 about point G. Point Q is the midpoint of segment EF. Let X be the circumcircle of polygon ABCDEFGH. Let V be the inversion of point Q in circle X. What is the length of segment NV? 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 11. Line R is the perpendicular bisector of segment AU. Point W is the reflection of point E across line R. Line H passes through points C and U. Line P passes through point E parallel to segment WA. Point O is the intersection of lines H and P. Find the length of segment FO. 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 V and circumradius 12. Let K be the reflection of point E across point B. Point W is the reflection of point A across point K. Let J be the circumcircle of polygon ABCDE. Let G be the inversion of point W in circle J. Point T is the midpoint of segment CG. 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 ABCDEFGH be a regular octagon with center N and circumradius 2. Line J is the angle bisector of angle DAG. Line S is the perpendicular bisector of segment EC. Line K is the reflection of line S across line J. Point Z is the reflection of point B across line K. Find the length of segment ZG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center L and circumradius 9. Let S be the reflection of point F across point E. Point J is obtained by rotating point S by angle pi/4 about point A. Point R is the midpoint of segment CA. Let I be the angle bisector of angle HRJ. Line U passes through points E and C. Let P be the reflection of line I across line U. Lines P and U intersect at point N. Find the distance between points N and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center P and circumradius 10. Line Q is the angle bisector of angle GHF. Let Z be the line through point K parallel to line Q. Point N is the reflection of point F across line Z. Let T be the midpoint of segment NI. Find the distance between points E and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center I and circumradius 8. Point Z is the reflection of point C across segment EA. Point N is the midpoint of segment EA. Let V be the midpoint of segment NB. 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]
Let ABCDEFGH be a regular octagon with center Q and circumradius 9. Point T is the midpoint of segment EB. Point Z lies on segment HD. Line P passes through point T perpendicular to segment HD. Point I is the reflection of point Z across line P. Find the measure of angle FHI, 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 12. Let M be the midpoint of segment LC. Point N is the reflection of point A across point E. Point X is the reflection of point M across segment HJ. Point R is the midpoint of segment XJ. Find the length of segment NR. 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 8. Line R passes through points I and B. Segment LE and line R intersect at point X. Point U is the reflection of point H across point X. Line N is the angle bisector of angle JUI. Point V is the reflection of point H across line N. What is the length of segment LV? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDE is a regular 5-gon with center U and circumradius 2. Let T be the line through points D and E. Let P be the rotation of point B by angle 3pi/4 around point U. Let N be the perpendicular bisector of segment AP. Line I is the reflection of line N across line T. Point O is the reflection of point B across line I. Point J is the reflection of point O across line N. What is the length of segment OJ? Express your answer as a decimal number rounded to 2 decimal places.

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

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[User]
Let ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 5. Let O be the line through points L and A. Point T is the midpoint of segment GC. Point Q is the reflection of point H across line O. Point W is the midpoint of segment TK. What is the distance between points W and Q? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDFGHIJKLM is a regular dodecagon with center E and circumradius 4. Point N is the midpoint of segment GB. Let R be the reflection of point A across point M. Line Q passes through point F perpendicular to segment LH. Point W is the reflection of point R across line Q. Find the length of segment NW. 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 U and circumradius 8. Point V is the midpoint of segment AU. Let K be the circumcircle of polygon ABCDEF. Let Y be the reflection of circle K through point D. Let W be the inversion of point F in circle Y. Point L is the reflection of point E across point V. Find the length of segment LW. 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 W and circumradius 10. Let V be the perpendicular bisector of segment EF. Let U be the line through point B perpendicular to line V. Point M lies on line U. Find the measure of angle ABM, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEFGH is a regular octagon with center Z and circumradius 4. Let T be the angle bisector of angle AHC. Line W passes through point F parallel to line T. Line L is the reflection of line W across line T. Point R is the reflection of point B across line L. Point J is the midpoint of segment DR. Find the distance between points R and J. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEF is a regular hexagon with center J and circumradius 1. Line M passes through points C and E. Let S be the reflection of point A across point B. Let L be the circumcircle of polygon ABCDEF. Point Q is the reflection of point B across line M. Let O be the reflection of circle L through point B. Point Y is the reflection of point J across point C. Let X be the inversion of point Y in circle O. Point T is the reflection of point X across segment SD. Find the length of segment QT. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEFGHIJKM is a regular dodecagon with center L and circumradius 2. Let T be the circumcircle of polygon ABCDEFGHIJKM. Point V is the reflection of point A across point I. Let X be the line through points E and V. Circle Z is the reflection of circle T across point B. Let W be the perpendicular bisector of segment MC. Point U is the reflection of point D across segment MC. Let R be the reflection of line X across line W. Point P is the reflection of point U across line W. Lines X and R intersect at point Y. Let Q be the inversion of point P in circle Z. Let N be the inversion of point Q in circle T. Find the distance between points Y and N. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ABCDEFG be a regular 7-gon with center S and circumradius 1. Point R is the reflection of point E across point S. Point H is the midpoint of segment RC. Let I be the midpoint of segment FG. Let Y be the reflection of point I across point E. What is the distance between points H and Y? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEFGHIJK is a regular 11-gon with center V and circumradius 2. Let X be the line through points J and F. Line S passes through points A and D. Let Z be the reflection of line S across line X. Point P is the reflection of point B across line Z. What is the distance between points A and P? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ABCDEFGHIJK be a regular 11-gon with center P and circumradius 9. Point N is the reflection of point D across point J. Let U be the line through points D and H. Point Y is the midpoint of segment IN. Let L be the line through point Y parallel to line U. Point T is the reflection of point J across line L. What is the distance between points T and D? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ABCDE be a regular 5-gon with center W and circumradius 8. Let T be the circumcircle of polygon ABCDE. Let G be the reflection of circle T through point A. Let Q be the reflection of point E across point C. Let J be the inversion of point Q in circle G. Find the length of segment JB. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEFGH is a regular octagon with center T and circumradius 7. Let L be the line through points H and G. Let Y be the circumcircle of polygon ABCDEFGH. Circle W is the reflection of circle Y across line L. Point V is the midpoint of segment CA. Point X is the midpoint of segment VB. Point Q is the inversion of point E with respect to circle W. Find the length of segment QX. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEF is a regular hexagon with center G and circumradius 10. Let H be the midpoint of segment EA. Let O be the angle bisector of angle BHA. Let U be the line through point H perpendicular to line O. Point I is the reflection of point B across line U. Find the distance between points I and B. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEFGH is a regular octagon with center R and circumradius 4. Line L is the angle bisector of angle RHD. Let Q be the angle bisector of angle HGF. Line I is the reflection of line Q across line L. Line W passes through point H parallel to line Q. Point M is the midpoint of segment DA. Lines W and I intersect at point O. Let N be the reflection of point F across point O. Point Z is the midpoint of segment MN. Find the measure of angle DBZ, in radians. 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 R and circumradius 10. Line Q passes through points B and H. Point Z is the reflection of point I across line Q. Let S be the midpoint of segment ZA. Point N is the reflection of point S across point G. What is the length of segment ND? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCD is a square with center O and circumradius 8. Let Z be the midpoint of segment DC. Line W is the perpendicular bisector of segment AO. Let M be the midpoint of segment CO. Point K is the reflection of point M across line W. Line V is the angle bisector of angle ZMA. Point Q is the reflection of point K across line V. Find the distance between points Q and Z. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ABCDEFGH be a regular octagon with center I and circumradius 6. Point J is the midpoint of segment GE. Line T is the perpendicular bisector of segment JB. Line K passes through point C perpendicular to line T. Point Z is the midpoint of segment HA. Point Y is the reflection of point Z across line K. What is the distance between points Y and H? 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 L and circumradius 11. Let S be the line through point F parallel to segment AE. Let Z be the midpoint of segment LA. Let M be the line through point Z perpendicular to line S. Point Q is the reflection of point C across line M. Find the distance between points Q and C. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ABCD be a square with center F and circumradius 2. Let K be the reflection of point C across point D. Let H be the circumcircle of polygon ABCD. Let X be the midpoint of segment AD. Let Z be the reflection of circle H through point K. Point L is the inversion of point X with respect to circle Z. Point V is the reflection of point K across point A. Point Q is the reflection of point L across point V. Find the distance between points A and Q. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 1. Let R be the perpendicular bisector of segment EA. Let S be the midpoint of segment GK. Point Q is the reflection of point S across line R. Find the length of segment IQ. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEFGHIJK is a regular 11-gon with center S and circumradius 5. Let W be the midpoint of segment HD. Let U be the midpoint of segment CW. Point T is the midpoint of segment AB. Point Y is the reflection of point E across point U. What is the length of segment YT? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEF is a regular hexagon with center Q and circumradius 8. Point Y is the midpoint of segment AF. Point L is the midpoint of segment BC. Point H is the midpoint of segment QY. Let R be the reflection of point A across point H. Point T is the reflection of point R across point A. What is the distance between points T and L? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEGHIJK is a regular 10-gon with center F and circumradius 8. Let N be the rotation of point K by angle 11pi/12 around point F. Point L is the reflection of point N across point D. Let U be the line through points L and G. Let R be the midpoint of segment HJ. Point V is the reflection of point N across line U. Find the length of segment VR. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEFG is a regular 7-gon with center L and circumradius 8. Let J be the midpoint of segment BD. Line Y is the perpendicular bisector of segment LC. Let P be the reflection of line Y through point J. Let X be the line through points D and G. Lines P and X intersect at point M. Point U is the reflection of point M across line Y. Find the distance between points U and D. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ABCDEF is a regular hexagon with center J and circumradius 4. Let U be the circumcircle of polygon ABCDEF. Circle I is the reflection of circle U across point E. Let K be the midpoint of segment JF. Let O be the inversion of point F in circle I. 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}