[User]
Let ABCDEFGHI be a regular 9-gon with center W and circumradius 4. Let Z be the midpoint of segment GW. Point L is the midpoint of segment ZE. Let T be the midpoint of segment AL. Let N be the reflection of point T across point G. What is the distance between points N and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center O and circumradius 1. Point G is the reflection of point C across point B. Let L be the perpendicular bisector of segment AD. Line I passes through point G perpendicular to line L. Point N is the reflection of point E across line I. Find the measure of angle END, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center R and circumradius 1. Point Z is the reflection of point D across point B. Let S be the circumcircle of polygon ABCDEFGHIJKL. Circle W is the reflection of circle S across point D. Let P be the inversion of point K in circle W. What is the length of segment PZ? 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 5. Let N be the line through points G and B. Line Z is the reflection of line N across point A. Point J is the reflection of point A across line Z. Let I be the reflection of point B across point J. Find the length of segment IJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center N and circumradius 12. Point K is the reflection of point A across point B. Let Y be the midpoint of segment AN. Point V is the midpoint of segment YK. Find the length of segment VD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFG is a regular hexagon with center A and circumradius 12. Let O be the midpoint of segment FD. Point Q is the reflection of point O across point B. Let M be the reflection of point E across point C. What is the length of segment MQ? 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 12. Point X is the midpoint of segment KB. Let Z be the angle bisector of angle HXD. Point Q is the reflection of point C across line Z. Let U be the midpoint of segment GQ. What is 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]
Let ABDEF be a regular 5-gon with center C and circumradius 10. Let Y be the circumcircle of polygon ABDEF. Point T is the midpoint of segment FE. Point V is the inversion of point T with respect to circle Y. Point W is the reflection of point V across point E. Find the distance between points T and W. 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 5. Let N be the midpoint of segment GD. Let Y be the line through point N perpendicular to segment NJ. Point P is the reflection of point H across line Y. What is the length of segment KP? 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 W and circumradius 3. Line Q passes through points G and I. Point L is the reflection of point A across line Q. Let K be the midpoint of segment LW. What is the distance between points F 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 ABCEFGH be a regular 7-gon with center D and circumradius 10. Line N passes through points G and F. Let P be the circumcircle of polygon ABCEFGH. Circle L is the reflection of circle P across line N. Let X be the inversion of point E in circle L. Find the distance between points X 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 ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 11. Let Q be the angle bisector of angle BHE. Point U is the midpoint of segment CB. Let V be the line through point D parallel to line Q. Let S be the reflection of point H across point U. Point R is the reflection of point L across segment CI. Point P is the reflection of point A across line V. What is the measure of angle SPR, 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 X and circumradius 4. Line I is the perpendicular bisector of segment FE. Line R is the reflection of line I across point G. Let H be the reflection of line R through point E. Point K is the reflection of point E across line H. Find the distance between points K and D. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center T and circumradius 10. Point H is obtained by rotating point T by angle pi/12 about point A. Let I be the midpoint of segment FH. Line M is the perpendicular bisector of segment TF. Point L is the reflection of point C across line M. Let W be the reflection of point I across point L. Let Z be the reflection of point W across point H. What is the length of segment ZE? 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 8. Point W is the midpoint of segment BF. Let S be the midpoint of segment WD. Let H be the circumcircle of polygon ABCDEG. Point O is the inversion of point S with respect to circle H. Find the length of segment AO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFGHI is a regular octagon with center C and circumradius 12. Let N be the line through point E perpendicular to segment FE. Let Z be the line through point F parallel to line N. Let O be the perpendicular bisector of segment FE. Point J is the reflection of point B across line Z. Point X is the reflection of point J across line O. Point R is the reflection of point X across point A. Find the length of segment RE. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center R and circumradius 9. Point Z is the reflection of point A across point D. Point O is the midpoint of segment CZ. Let Y be the perpendicular bisector of segment RZ. Point H is the reflection of point O across line Y. Find the length of segment HA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center I and circumradius 12. Let E be the rotation of point D by angle 11pi/12 around point I. Point V is the reflection of point B across segment EC. Point L is the midpoint of segment EV. What is the distance between points L and E? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center I and circumradius 3. Point Z is the reflection of point H across point A. Let R be the midpoint of segment DC. Line Q is the angle bisector of angle RBE. Point O is the reflection of point R across line Q. Point P is the reflection of point G across point O. What is the distance between points P and Z? 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 9. Point Y is the midpoint of segment DC. Let N be the midpoint of segment CI. Let W be the line through point I perpendicular to segment AF. Point J is the reflection of point Y across line W. What is the length of segment NJ? 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 1. Line I passes through point E perpendicular to segment BD. Point R is the reflection of point A across line I. Let Y be the midpoint of segment BD. Find the distance between points R and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center M and circumradius 5. Line X is the angle bisector of angle CEM. Point J is the reflection of point M across line X. Let W be the line through points A and J. Point S is the reflection of point M across line W. Find the distance between points J and S. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center R and circumradius 4. Let Q be the rotation of point H by angle 5pi/12 around point A. Line O is the perpendicular bisector of segment EQ. Point Y is the reflection of point D across line O. Let X be the perpendicular bisector of segment YA. Point T is the reflection of point J across line X. Find the distance between points T and Y. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center R and circumradius 2. Let X be the midpoint of segment GA. Let O be the perpendicular bisector of segment HD. Point U is the reflection of point X across line O. Find the length of segment CU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFHIJKLM is a regular dodecagon with center G and circumradius 4. Line O is the angle bisector of angle FIH. Point Q is the reflection of point D across line O. Let T be the midpoint of segment JQ. What is the measure of angle TJF, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center P and circumradius 12. Let Y be the angle bisector of angle DBE. Let T be the rotation of point G by angle 2pi/3 around point H. Let Q be the line through point P perpendicular to segment GB. Let W be the reflection of line Q across line Y. Point Z is the reflection of point G across line W. What is the distance between points T 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 ABDEFGHIJKL be a regular 11-gon with center C and circumradius 2. Line O is the angle bisector of angle DGE. Let S be the circumcircle of polygon ABDEFGHIJKL. Circle Y is the reflection of circle S across line O. Let R be the inversion of point K in circle Y. Find the distance between points R and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIKLM is a regular dodecagon with center J and circumradius 4. Let U be the perpendicular bisector of segment CJ. Let P be the line through point L parallel to line U. Segment BI and line P intersect at point T. What is the measure of angle ITK, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center P and circumradius 1. Line W is the perpendicular bisector of segment CJ. Line T passes through points A and I. Line V is the reflection of line W across point I. Point U is the intersection of lines V and T. Let X be the midpoint of segment DA. What is the length of segment XU? 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 5. Line S passes through point F parallel to segment ED. Let X be the midpoint of segment ED. Point N is the reflection of point X across line S. What is the length of segment EN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center X and circumradius 11. Let N be the angle bisector of angle DXG. Let V be the line through point D parallel to line N. Let J be the reflection of point A across point B. Point L is the reflection of point J across line V. Find the length of segment LG. 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 8. Let L be the angle bisector of angle AJE. Let W be the line through point B parallel to line L. Let G be the circumcircle of polygon ABCDEF. Let I be the reflection of circle G across line W. Let M be the inversion of point J in circle I. Find the distance between points M and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center O and circumradius 4. Let X be the line through points B and C. Line I passes through point C perpendicular to line X. Line P passes through point D parallel to line I. Let R be the circumcircle of polygon ABCDEFGH. Let W be the reflection of line P across line I. Circle N is the reflection of circle R across line W. Let U be the inversion of point C in circle N. Find the length of segment UF. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center I and circumradius 2. Point J is obtained by rotating point G by angle 5pi/12 about point A. Point X is the reflection of point D across point H. Line U passes through point I perpendicular to segment BJ. Point W is the reflection of point B across line U. Point T is the reflection of point W across segment EJ. Find the length of segment TX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFGH is a regular 7-gon with center D and circumradius 5. Let R be the reflection of point G across point D. Let N be the line through points R and C. Point I lies on line N. Line U is the perpendicular bisector of segment AI. Point X is the reflection of point H across line U. Find the length of segment XI. 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 3. Line V is the angle bisector of angle JLD. Line W passes through point E perpendicular to line V. Point Z is the reflection of point A across line W. Let T be the midpoint of segment KZ. What is 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]
BCDEFGHIJK is a regular 10-gon with center A and circumradius 6. Let V be the midpoint of segment GF. Line W passes through points D and H. Line U passes through point V perpendicular to line W. Point S is the reflection of point D across line U. What is the distance between points A and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 3. Let X be the angle bisector of angle BMD. Line O passes through point G parallel to line X. Point Q is the reflection of point B across line O. Point T is the reflection of point Q across line X. Find the distance between points G 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 ABCEFGHI be a regular octagon with center D and circumradius 2. Point N is the midpoint of segment FB. Let R be the line through point G parallel to segment AD. Line O passes through points C and E. Lines R and O intersect at point X. Find the length of segment NX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center O and circumradius 12. Let Y be the angle bisector of angle HOE. Line S is the reflection of line Y across point H. Point W is the reflection of point I across line S. Let N be the reflection of point L across point A. What is the length of segment NW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFHIJ be a regular 9-gon with center G and circumradius 1. Point U is the reflection of point C across point B. Line O is the perpendicular bisector of segment HC. Line M is the perpendicular bisector of segment UD. Let S be the reflection of line M across line O. Line S and segment UD intersect at point V. What is the length of segment HV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center R and circumradius 7. Let W be the perpendicular bisector of segment JR. Line K is the angle bisector of angle JGR. Line T is the reflection of line W across line K. Let L be the reflection of line T through point I. Line U is the reflection of line L across line W. Point Y is the reflection of point A across line U. Point X is the reflection of point Y across line L. Point N is the reflection of point X across line U. What is the length of segment NJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center M and circumradius 8. Let N be the perpendicular bisector of segment CB. Line O passes through point A parallel to line N. Let Y be the midpoint of segment AD. Point W is the reflection of point Y across point B. Point T is the reflection of point D across line O. Find the distance between points T 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 BCDEFGHIJKLM be a regular dodecagon with center A and circumradius 7. Point Z is the midpoint of segment GC. Let U be the midpoint of segment GZ. Let V be the midpoint of segment UK. Find the length of segment BV. 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 8. Let K be the circumcircle of polygon ABCDEFG. Circle P is the reflection of circle K across point A. Point S is the midpoint of segment CT. Point N is the inversion of point E with respect to circle P. Let I be the line through points N and S. Let J be the reflection of line I through point B. Point O is the midpoint of segment TF. Point V is the reflection of point O across line J. 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 ABCDEFGHIJ be a regular 10-gon with center Q and circumradius 9. Point Z is the reflection of point C across point J. Point T is the midpoint of segment ZQ. Point S is the midpoint of segment QD. Find the length of segment ST. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center W and circumradius 4. Line X passes through points D and H. Point P is obtained by rotating point K by angle 7pi/12 about point E. Let Q be the angle bisector of angle IWP. Line S passes through point C parallel to line X. Let Z be the reflection of line Q across line S. Point R is the intersection of lines Z and X. Find the distance between points F and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center P and circumradius 12. Point V is the reflection of point D across point G. Point K is the reflection of point G across point V. Let M be the midpoint of segment ED. Let O be the midpoint of segment PD. Point T is the reflection of point M across segment FK. Find 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]
Let ABCDEFGHIJKL be a regular dodecagon with center S and circumradius 8. Let R be the line through points H and C. Point P is the reflection of point F across line R. Let V be the angle bisector of angle EHI. Point M is the reflection of point B across line V. What is the measure of angle MPA, 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 H and circumradius 6. Point N is obtained by rotating point B by angle 5pi/6 about point D. Let I be the circumcircle of polygon ABCDEF. Let Q be the reflection of circle I through point D. Let P be the inversion of point N in circle Q. Let Y be the line through points F and D. Point V is the reflection of point P across line Y. Find the length of segment BV. 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 Z and circumradius 4. Line W is the perpendicular bisector of segment BA. Line R is the perpendicular bisector of segment JB. Line M is the reflection of line W across line R. Point L is the intersection of segment KG and line M. What is the length of segment FL? 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 6. Let W be the midpoint of segment FX. Let Q be the midpoint of segment EW. Let K be the line through points C and Q. Point L is the reflection of point B across line K. What is the distance between points B and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center W and circumradius 3. Let Q be the reflection of point W across point C. Let G be the rotation of point E by angle 7pi/12 around point W. Let R be the rotation of point C by angle pi/4 around point D. Let U be the vector from point F to point D. Let Z be point R translated by vector U. Line V is the angle bisector of angle EZR. Line L passes through point D parallel to line V. Point P is the reflection of point W across segment QG. Let M be the line through points F and P. Lines M and L intersect at point T. What is 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]
BH = 11. Let ACDEFG be a regular hexagon with center H and circumradius 12. Let K be the rotation of point C by angle pi around point F. Let W be the reflection of point G across point C. Point M is the reflection of point K across point D. Let N be the midpoint of segment WC. What is the distance between points M 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 ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 5. Line W is the angle bisector of angle GIF. Let M be the circumcircle of polygon ABCDEFGHIJKL. Circle T is the reflection of circle M across line W. Let O be the inversion of point K in circle T. 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 ABCDEFGHIJL be a regular 11-gon with center K and circumradius 10. Line N is the perpendicular bisector of segment KB. Point W is the reflection of point D across line N. Line R is the angle bisector of angle LWD. Line Z is the reflection of line N across line R. Let U be the reflection of line N across line Z. Point X is the reflection of point G across line U. Find the distance between points X and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 12. Let P be the line through points C and F. Point N is the reflection of point H across point J. Line U is the perpendicular bisector of segment HE. Let Y be the reflection of line P across line U. Point R is the reflection of point F across line Y. Line V passes through point N perpendicular to line P. Let Q be the line through point S parallel to line V. Point T is the intersection of lines Q and P. Point M is the midpoint of segment TC. Find the distance between points M and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center V and circumradius 7. Let U be the line through points E and D. Let O be the reflection of line U through point A. Line P is the reflection of line U across line O. Point R is the reflection of point B across point F. Point S is the reflection of point D across line P. Find the length of segment SR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEF be a regular 5-gon with center C and circumradius 1. Point L is the reflection of point A across point C. Point W is the reflection of point A across point L. Point Q is the midpoint of segment WC. Let K be the midpoint of segment WB. What is the distance between points K and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center M and circumradius 7. Line N is the angle bisector of angle BCF. Let X be the reflection of line N through point E. Let R be the rotation of point M by angle 5pi/12 around point C. Point Z is the reflection of point R across line X. Point O is the midpoint of segment GI. Let P be the midpoint of segment OM. What is 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]
ABCDEF is a regular hexagon with center Y and circumradius 6. Let G be the line through points E and F. Point Q is the reflection of point D across line G. Let M be the midpoint of segment QE. Let Z be the reflection of point Y across point M. What is the distance between points Z 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 ABCDEF be a regular hexagon with center N and circumradius 9. Let Y be the perpendicular bisector of segment CB. Line J is the reflection of line Y across point A. Point I is the reflection of point C across line J. Point Z is the midpoint of segment DF. Point T is the reflection of point I across point Z. Find the length of segment TA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGI is a regular octagon with center H and circumradius 8. Let Q be the perpendicular bisector of segment CB. Let X be the angle bisector of angle FGD. Line P is the reflection of line Q across line X. Let V be the line through point G perpendicular to line P. Let M be the reflection of line V through point H. Point Y is the reflection of point G across line M. Point L is the midpoint of segment YA. Find the length of segment LG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center W and circumradius 6. Line T is the perpendicular bisector of segment EJ. Let S be the line through points C and E. Let M be the reflection of line T across line S. Point V is the intersection of lines S and M. What is the distance between points C and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center O and circumradius 8. Let W be the rotation of point A by angle pi/6 around point D. Let X be the line through points O and A. Point P is the reflection of point W across line X. Point Q is the reflection of point C across segment AP. Find the distance between points Q and B. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center K and circumradius 3. Let V be the angle bisector of angle KAC. Point L is the reflection of point K across line V. Let P be the midpoint of segment LA. What is the distance between points B and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFHIJK is a regular 10-gon with center G and circumradius 2. Point N is obtained by rotating point K by angle 2pi/3 about point D. Point P is obtained by rotating point H by angle 5pi/12 about point K. Let T be the perpendicular bisector of segment GA. Line L passes through point P parallel to line T. Point O is the reflection of point F across line T. Point W is the reflection of point N across line L. Find the length of segment WO. 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 11. Point V is the reflection of point A across segment DB. Let M be the midpoint of segment KI. Let W be the reflection of point G across point V. What is the length of segment MW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center I and circumradius 9. Let P be the rotation of point E by angle pi/3 around point I. Point J is the midpoint of segment AB. Point Z is the midpoint of segment BP. Let X be the reflection of point Z across point A. Point G is the reflection of point X across segment BP. Find the distance between points J and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center M and circumradius 1. Let O be the perpendicular bisector of segment DM. Line H passes through point D perpendicular to line O. Point N is the reflection of point C across segment ED. Line Q passes through point D perpendicular to line H. Let Z be the line through points N and M. Let J be the reflection of line Z across line Q. Point Y is the intersection of line J and segment ED. What is the distance between points B and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center I and circumradius 8. Point O is obtained by rotating point D by angle pi/6 about point F. Line P is the angle bisector of angle OBC. Point Y is the reflection of point O across line P. Point W is the reflection of point Y across segment OD. What is the distance between points D and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center N and circumradius 8. Line V passes through points H and C. Point J lies on line V. Line Z passes through points E and J. Point I is the reflection of point A across line Z. What is the distance between points E 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 P and circumradius 2. Line R is the angle bisector of angle PDH. Point W is the reflection of point G across point F. Point Z is the reflection of point B across line R. Point V is the reflection of point Z across point W. What is the distance between points V and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center Y and circumradius 9. Let M be the line through point C perpendicular to segment FB. Let I be the reflection of line M through point F. Point U is the reflection of point E across line I. What is the measure of angle HEU, 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 P and circumradius 5. Let O be the midpoint of segment EP. Let I be the reflection of point O across point P. Point H is the midpoint of segment PC. What is the distance between points I 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 ABCDEFGHIJ be a regular 10-gon with center S and circumradius 4. Let U be the midpoint of segment JC. Point L is obtained by rotating point F by angle 3pi/4 about point H. Let Q be the perpendicular bisector of segment UF. Point Y is the reflection of point C across line Q. Line N is the angle bisector of angle CEY. Point T is the reflection of point U across line N. Find the length of segment TL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEFGHI be a regular octagon with center D and circumradius 5. Point U is the reflection of point I across point H. Let X be the angle bisector of angle EDA. Let O be the reflection of line X through point U. Let M be the line through point U parallel to line O. Line M and segment AC intersect at point J. Find the length of segment DJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEGHIJ be a regular 9-gon with center F and circumradius 8. Point R is the reflection of point E across segment JB. Let Q be the line through point I perpendicular to segment JB. Point O is the reflection of point F across line Q. What is the length of segment OR? 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 V and circumradius 3. Let M be the midpoint of segment KH. Let Z be the line through points A and F. Point U lies on line Z. Find the measure of angle UFM, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center T and circumradius 6. Let Y be the circumcircle of polygon ABCDE. Let N be the angle bisector of angle DBE. Let M be the reflection of circle Y through point D. Line Z passes through point B perpendicular to line N. Let X be the inversion of point C in circle M. Point U is the reflection of point X across line Z. What is the length of segment CU? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center T and circumradius 1. Point O is the midpoint of segment BT. Let W be the perpendicular bisector of segment CD. Point Y is the reflection of point O across line W. What is the length of segment YO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJ be a regular 10-gon with center Y and circumradius 2. Point O is the reflection of point H across point A. Point V is the midpoint of segment OF. Point Z is the midpoint of segment VG. What is the length of segment ZO? 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 M and circumradius 12. Let S be the vector from point J to point I. Point L is the reflection of point H across point A. Point U is the translation of point G by vector S. Let K be the line through points U and M. Line R passes through point H perpendicular to line K. Let T be the line through point G parallel to segment LB. Let V be the midpoint of segment LB. Line Y is the reflection of line R across line T. Point Z is the reflection of point V across line K. Lines T and Y intersect at point N. What is the distance between points Z 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 ABCDEFGHIJ be a regular 10-gon with center L and circumradius 5. Point Q is the reflection of point J across point D. Line T is the angle bisector of angle BQI. Line K passes through point E perpendicular to line T. Point M is the reflection of point A across line K. What is the length of segment MC? 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 11. Let J be the reflection of point A across point D. Let V be the circumcircle of polygon ABCD. Let S be the inversion of point J in circle V. Let R be the reflection of point S across point C. Let X be the midpoint of segment CJ. Let L be the inversion of point R in circle V. Find the length of segment XL. 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 5. Point Y is on segment GE. Point N is the reflection of point H across segment CY. Find 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]
ABCDEFG is a regular 7-gon with center U and circumradius 7. Point X is the reflection of point D across point F. Line Q is the angle bisector of angle FEB. Line W passes through points X and B. Point I is on line W. Line Z is the reflection of line Q across point I. Point P is the reflection of point X across line Z. Find the measure of angle FXP, 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 V and circumradius 6. Line Q passes through points F and B. Let M be the line through point C parallel to segment VE. Lines Q and M intersect at point H. What is the measure of angle ADH, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center R and circumradius 5. Let W be the perpendicular bisector of segment AI. Let V be the line through point F perpendicular to line W. Line M is the reflection of line V across point H. Point T is the reflection of point E across line M. What is the distance between points T 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 ABCDE be a regular 5-gon with center U and circumradius 6. Let I be the reflection of point B across point U. Let M be the midpoint of segment UB. Point L is the midpoint of segment EU. Let Y be the reflection of point L across point I. What is 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]
Let ABCDEFGH be a regular octagon with center O and circumradius 6. Let W be the angle bisector of angle EHB. Line X passes through point E parallel to line W. Let Z be the line through point D parallel to segment BC. Lines X and Z intersect at point T. Find the measure of angle BDT, 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 ABCDEFGHIJLM be a regular dodecagon with center K and circumradius 7. Let Q be the angle bisector of angle EAG. Line S is the perpendicular bisector of segment KC. Point T is the reflection of point M across line S. Point W is the reflection of point T across line Q. What is the measure of angle WTA, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center K and circumradius 8. Line L passes through points A and C. Let G be the circumcircle of polygon ABCDE. Let W be the rotation of point B by angle 5pi/12 around point E. Point Y is the reflection of point W across line L. Point J is the reflection of point E across line L. Let T be the reflection of circle G across line L. Point Q is the inversion of point Y with respect to circle T. Find the distance between points J and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center P and circumradius 8. Let S be the midpoint of segment PE. Let J be the midpoint of segment EC. Point I is the midpoint of segment SH. Find the length of segment JI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center O and circumradius 4. Let R be the perpendicular bisector of segment AB. Let J be the angle bisector of angle ADH. Line T passes through point B parallel to line J. Point L is the reflection of point C across line J. Let X be the reflection of line T across line R. Line U is the reflection of line R across line T. Let Y be the line through point L parallel to line U. Lines X and Y intersect at point N. 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]
Let ABCDE be a regular 5-gon with center J and circumradius 8. Let F be the rotation of point E by angle pi/3 around point C. Point O is obtained by rotating point F by angle 7pi/12 about point J. Line V is the angle bisector of angle OAF. Let N be the line through point O perpendicular to line V. Let L be the midpoint of segment AC. Point M is the reflection of point D across line N. Find the length of segment ML. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center G and circumradius 3. Let M be the rotation of point G by angle 5pi/12 around point D. Let Q be the rotation of point E by angle 7pi/12 around point A. Point Z is the reflection of point A across point D. Line P is the perpendicular bisector of segment QD. Let L be the angle bisector of angle MGD. Line Y is the reflection of line L across line P. Let J be the perpendicular bisector of segment GE. Let H be the reflection of line P across line J. Point R is the intersection of lines H and Y. What is the length of segment RZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEGHI be a regular octagon with center F and circumradius 6. Line Y passes through points H and B. Point N is the reflection of point F across point A. Line O is the reflection of line Y across point N. Point K is the reflection of point D across line O. What is the length of segment NK? 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 4. Let J be the reflection of point D across point B. Let W be the reflection of point E across point F. Let R be the midpoint of segment JH. Let S be the reflection of point R across point W. What is the length of segment RS? 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 8. Point Y is the reflection of point A across point I. Let P be the circumcircle of polygon ABCDEGHIJKLM. Point R is the midpoint of segment GY. Let Z be the inversion of point R in circle P. What is the distance between points Z and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJK is a regular 11-gon with center S and circumradius 11. Point V is the reflection of point H across point G. Line U passes through points I and C. Line O is the perpendicular bisector of segment HI. Line M is the reflection of line O across point G. Point X is the intersection of lines U and M. What is the distance between points V and X? 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 5. Let R be the angle bisector of angle HAG. Let X be the midpoint of segment IE. Point S is the reflection of point X across point E. Point U is the reflection of point H across line R. What is 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 ABCD be a square with center Y and circumradius 3. Let T be the perpendicular bisector of segment CB. Let W be the reflection of line T through point A. Point V is the midpoint of segment YC. Let F be the midpoint of segment DA. Point Z is the reflection of point F across line W. What is 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 ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 6. Line Q is the angle bisector of angle CJY. Let W be the reflection of point H across point F. Point X is the reflection of point E across line Q. Point T is the reflection of point X across point W. Point M is the reflection of point W across segment BT. What is the length of segment IM? 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 H and circumradius 6. Let O be the reflection of point G across point B. Point R is the midpoint of segment GD. Let Y be the perpendicular bisector of segment RO. Point X is the reflection of point G across line Y. Find the length of segment EX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center M and circumradius 11. Let S be the angle bisector of angle DMA. Let V be the perpendicular bisector of segment EA. Let I be the reflection of line V across line S. Line Z passes through point D parallel to line S. Lines I and Z intersect at point L. What is the measure of angle ELM, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center X and circumradius 6. Point N is the reflection of point B across point J. Let M be the line through point X perpendicular to segment XN. Let V be the reflection of line M through point C. Let L be the reflection of line V across line M. Line T is the reflection of line L across point B. Point O is the reflection of point E across line T. Find the length of segment OG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJ be a regular 10-gon with center L and circumradius 6. Let W be the angle bisector of angle JEI. Let O be the perpendicular bisector of segment JC. Let M be the reflection of line O through point H. Point Q is the reflection of point D across line M. Point X is the reflection of point Q across line W. What is the length of segment XI? 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 5. Line H passes through points B and A. Point W is on line H. Let U be the midpoint of segment BE. Find the measure of angle WBU, 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 T and circumradius 6. Line I passes through points T and E. Line V is the reflection of line I across point D. Point U is obtained by rotating point D by angle pi about point E. Let X be the midpoint of segment TD. Let G be the line through points E and X. Lines V and G intersect at point Y. Find the length of segment UY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFGHIJ is a regular 9-gon with center A and circumradius 2. Let Y be the reflection of point G across point J. Let P be the perpendicular bisector of segment DY. Let Z be the reflection of line P through point B. Point R is the intersection of segment DY and line Z. What is the distance between points R and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFG be a regular 7-gon with center O and circumradius 7. Let I be the circumcircle of polygon ABCDEFG. Let T be the midpoint of segment ED. Point J is the midpoint of segment CB. Circle H is the reflection of circle I across point C. Point X is the inversion of point T with respect to circle H. Find the distance between points J and X. 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 11. Let S be the reflection of point A across point G. Line P passes through points B and E. Let R be the midpoint of segment FS. Line M passes through point E perpendicular to line P. Let U be the reflection of line M through point F. Line X passes through points E and R. Lines X and U intersect at point J. What is the distance between points J and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGIJK is a regular 10-gon with center H and circumradius 5. Point W is the reflection of point H across segment FB. Let Y be the reflection of point W across point B. Let R be the midpoint of segment FB. Point V is the midpoint of segment RY. Find the distance between points G 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 ABCDEFHIJK be a regular 10-gon with center G and circumradius 3. Let P be the line through points J and E. Point L is the reflection of point E across point K. Line T passes through point L parallel to line P. Let W be the reflection of point A across point F. Segment AB and line T intersect at point X. Find the measure of angle WLX, 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 P and circumradius 1. Let R be the midpoint of segment CA. Let W be the line through points D and F. Line S is the reflection of line W across point B. Point M is the reflection of point R across line S. Find the distance between points E and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center Q and circumradius 12. Let P be the angle bisector of angle GEQ. Let T be the line through point Q perpendicular to line P. Let V be the midpoint of segment QD. Point M is the reflection of point V across line T. What is the distance between points M and B? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center Q and circumradius 5. Point H is the midpoint of segment DF. Let S be the perpendicular bisector of segment HQ. Let W be the line through point B parallel to line S. Point M is the reflection of point A across line W. What is the distance between points M 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 ABCDEFGH be a regular octagon with center Y and circumradius 2. Point W is the reflection of point Y across point A. Let Z be the rotation of point A by angle 5pi/12 around point B. Let J be the circumcircle of polygon ABCDEFGH. Circle T is the reflection of circle J across point B. Point M is the inversion of point G with respect to circle T. Point X is the intersection of segments WE and ZH. Find the length of segment XM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center M and circumradius 11. Point S is obtained by rotating point E by angle pi about point B. Let Q be the midpoint of segment EM. Point W is the reflection of point B across segment AQ. Point G is the reflection of point S across segment QC. What is the distance between points G and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center L and circumradius 12. Let Y be the rotation of point A by angle 2pi/3 around point L. Let V be the rotation of point L by angle pi/3 around point Y. Let Z be the line through point V parallel to segment YB. Let N be the line through point L perpendicular to line Z. Point P is the reflection of point A across line N. Find the distance between points P and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center S and circumradius 4. Let M be the midpoint of segment SA. Let L be the midpoint of segment SF. Point Z is the reflection of point M across point L. Point H is the reflection of point M across segment CA. What is the distance between points Z and H? 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 Z and circumradius 3. Let R be the circumcircle of polygon ABCD. Point J is obtained by rotating point Z by angle pi/6 about point B. Line H is the angle bisector of angle ACJ. Let Q be the reflection of circle R through point A. Point T is the inversion of point C with respect to circle Q. Line W passes through point T perpendicular to line H. Point N is the reflection of point D across line W. Find the distance between points Z and N. 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. Point M is the reflection of point D across segment HG. Point X is the midpoint of segment HG. Let Y be the midpoint of segment XC. What is the length of segment MY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHI is a regular 9-gon with center Q and circumradius 8. Let R be the circumcircle of polygon ABCDEFGHI. Point K is obtained by rotating point E by angle 7pi/12 about point A. Point X is the midpoint of segment KQ. Let Y be the reflection of circle R through point A. Point Z is the inversion of point X with respect to circle R. Point J is the inversion of point Z with respect to circle Y. Find the length of segment IJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center K and circumradius 11. Let Q be the line through points H and B. Point Z is on line Q. Let O be the angle bisector of angle ZHC. Point P is the reflection of point G across line O. Line V passes through point D perpendicular to line O. Point X is the reflection of point P across segment FD. Point W is the reflection of point P across line V. What is the length of segment XW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJK be a regular 11-gon with center W and circumradius 12. Point Y is the midpoint of segment CJ. Let M be the perpendicular bisector of segment KI. Point N is the reflection of point Y across line M. Let R be the midpoint of segment YC. Point P is the midpoint of segment NC. What is the distance between points P and R? 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 Q and circumradius 8. Let T be the rotation of point D by angle pi/12 around point B. Point J is the midpoint of segment QD. Let Y be the perpendicular bisector of segment CT. Let Z be the reflection of point A across point C. Line X passes through point J perpendicular to line Y. Let M be the reflection of point Z across point J. Point V is the reflection of point M across line X. Find the length of segment CV. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFG is a regular hexagon with center A and circumradius 5. Let I be the perpendicular bisector of segment EG. Let N be the line through point C perpendicular to line I. Let S be the circumcircle of polygon BCDEFG. Let H be the reflection of circle S across line N. Point O is the inversion of point A with respect to circle H. 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]
ABCDEFGH is a regular octagon with center X and circumradius 7. Let Y be the vector from point D to point C. Let J be the midpoint of segment CB. Let P be the line through points E and A. Let I be point J translated by vector Y. Line T passes through points C and I. Let K be the reflection of line T across line P. Let N be the reflection of point E across point J. Let S be the reflection of line K through point I. Point O is the reflection of point A across line S. Let Q be the reflection of point N across point X. Find the distance between points Q and O. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGI is a regular octagon with center H and circumradius 9. Line L is the perpendicular bisector of segment FA. Let W be the line through point B perpendicular to line L. Point P is the reflection of point I across line W. Point Q is the reflection of point P across line L. What is the distance between points B and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center X and circumradius 1. Let N be the line through points H and D. Let J be the line through point E perpendicular to line N. Line Q passes through point G parallel to line J. Point Y is the reflection of point C across line Q. What is the distance between points B and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ACDEFGHIJKLM is a regular dodecagon with center B and circumradius 11. Let U be the angle bisector of angle KMH. Line R passes through point H parallel to line U. Point S is the reflection of point K across line R. Point Y is the midpoint of segment JS. Find the distance between points Y and M. 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 9. Let T be the line through points G and A. Line L passes through point W parallel to line T. Line L and segment FE intersect at point P. What is the distance between points P and A? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center J and circumradius 11. Point L is the reflection of point A across point B. Point G is the reflection of point C across point L. Point X is the midpoint of segment AC. Point W is the reflection of point X across point G. What is the length of segment LW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BCDEF be a regular 5-gon with center A and circumradius 4. Point M is the reflection of point A across segment FE. Let O be the midpoint of segment FE. Let P be the angle bisector of angle OBM. Point X is the reflection of point A across line P. Find the length of segment XM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center N and circumradius 4. Point V is the reflection of point B across point A. Let R be the line through points C and V. Point G is the midpoint of segment VF. Point Q is the midpoint of segment VG. Point U is the reflection of point D across line R. Let L be the midpoint of segment QU. What is the length of segment LF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center H and circumradius 9. Let T be the angle bisector of angle DCE. Point N is the reflection of point D across line T. Let L be the midpoint of segment CE. Find the measure of angle DNL, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ABDEFGHI be a regular octagon with center C and circumradius 5. Let M be the line through points G and I. Let U be the line through point F parallel to line M. Line K is the reflection of line U across point E. Line O passes through point C perpendicular to line U. Let V be the reflection of line O through point I. Lines V and K intersect at point Z. What is the length of segment ZB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center P and circumradius 12. Line O is the angle bisector of angle CBE. Line Z is the reflection of line O across point A. Point G lies on line Z at distance 9 from point F. Line L is the reflection of line O across point F. Let W be the reflection of line L across line O. Let S be the reflection of line O across line W. Point N is the reflection of point G across line S. What is the distance between points N 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 ABCDE be a regular 5-gon with center H and circumradius 1. Line X is the perpendicular bisector of segment BA. Let S be the reflection of line X through point A. Line L is the reflection of line S across point C. Point Z is the midpoint of segment BA. Point G is the reflection of point E across point Z. Point M is the reflection of point A across line L. Find the distance between points G and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EA = 3. BCDFGH is a regular hexagon with center A and circumradius 12. Let L be the line through point H parallel to segment GA. Point Z is the reflection of point D across line L. Point R is the reflection of point Z across segment GA. Find the length of segment GR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEGHIJKL be a regular 11-gon with center F and circumradius 2. Line Z is the perpendicular bisector of segment CE. Let M be the line through point I parallel to line Z. Line P passes through point H parallel to segment CE. Point T is the intersection of lines P and M. What is the distance between points E and T? 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 Q and circumradius 8. Let V be the line through points A and B. Point R is the reflection of point G across line V. Line P is the perpendicular bisector of segment HQ. Point N is the reflection of point F across line P. What is 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]
Let ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 8. Line O passes through points H and E. Let U be the angle bisector of angle BAK. Line X is the reflection of line O across line U. Point N is the reflection of point H across line X. Find the length of segment NE. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center Z and circumradius 3. Point K is obtained by rotating point E by angle 3pi/4 about point C. Let M be the line through points B and C. Let N be the perpendicular bisector of segment DK. Let I be the reflection of line N through point E. Lines M and I intersect at point J. Find the distance between points J and D. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDFGHI be a regular octagon with center E and circumradius 7. Let O be the midpoint of segment HE. Line L is the angle bisector of angle OIB. Let J be the circumcircle of polygon ABCDFGHI. Circle T is the reflection of circle J across line L. Let Z be the inversion of point F in circle T. Find the distance between points Z and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center S and circumradius 3. Point F is the reflection of point B across point D. Let G be the reflection of point S across point D. Let U be the circumcircle of polygon ABCD. Let N be the inversion of point F in circle U. What is 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]
Let ABCDEFGHIJKL be a regular dodecagon with center M and circumradius 9. Let Q be the perpendicular bisector of segment CE. Let Y be the reflection of point C across point G. Let N be the reflection of line Q through point K. Line T is the reflection of line N across line Q. Point U is the reflection of point D across line T. Let Z be the angle bisector of angle KUY. Point V is the reflection of point H across line Z. What is the distance between points C and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center Q and circumradius 8. Let H be the midpoint of segment ED. Line K passes through points A and D. Let N be the line through point E perpendicular to line K. Let S be the reflection of point H across point E. Let W be the perpendicular bisector of segment HG. Let Y be the circumcircle of polygon ABCDEFG. Point P is the reflection of point S across line W. Circle I is the reflection of circle Y across line N. Line L passes through points F and P. Let U be the reflection of circle I across line L. Let R be the inversion of point Q in circle U. Find the distance between points R and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center R and circumradius 3. Let T be the midpoint of segment DA. Let N be the angle bisector of angle FET. Point I is the reflection of point D across line N. Point P is the reflection of point I across segment DA. What is the distance between points P and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCE is a square with center D and circumradius 7. Point P lies on segment CA. Line H is the perpendicular bisector of segment BP. Point S is the reflection of point D across line H. Line O passes through point S parallel to line H. Point Q is the reflection of point P across line O. What is the distance between points Q and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center P and circumradius 4. Let K be the rotation of point C by angle pi around point G. Point V is the midpoint of segment KG. Point N is the reflection of point P across point V. Line O passes through point N perpendicular to segment HK. Let I be the line through point P perpendicular to line O. Point T is the reflection of point F across line I. Let R be the midpoint of segment HK. Find the distance between points R and T. 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 3. Point U is on segment BA. Let G be the line through point F perpendicular to segment BA. Point Y is the reflection of point U across point V. Point M is the reflection of point U across line G. What is the distance between points Y and M? 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 T and circumradius 8. Point X is obtained by rotating point C by angle pi/6 about point F. Line W is the perpendicular bisector of segment AF. Line P is the reflection of line W across point C. Line K passes through point D perpendicular to line P. Point Y is the reflection of point E across line K. What is the length of segment YX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJK be a regular 11-gon with center Q and circumradius 9. Let V be the line through points B and J. Let L be the rotation of point D by angle 5pi/6 around point C. Line N passes through points L and K. Let O be the line through points A and H. Lines V and N intersect at point P. Point M is the intersection of lines N and O. What is the distance between points M and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center K and circumradius 7. Let Z be the line through points C and F. Let O be the circumcircle of polygon ABCDEF. Line W passes through point D parallel to line Z. Circle P is the reflection of circle O across line W. Point Y is the inversion of point F with respect to circle P. Find the distance between points Y and C. 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 1. Let X be the midpoint of segment MH. Let J be the midpoint of segment XA. Point Y is the reflection of point D across point E. Let Q be the midpoint of segment EY. Find the distance between points Q and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 5. Line Y is the angle bisector of angle MBH. Point X is the midpoint of segment JH. Point S is the reflection of point K across line Y. Point T is the midpoint of segment SX. Find the length of segment TL. 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 F and circumradius 12. Point U is the midpoint of segment CD. Point L is the reflection of point U across point B. Let I be the rotation of point L by angle pi around point U. Point E is the midpoint of segment LF. Point K is the reflection of point I across point A. Let P be the circumcircle of polygon ABCD. Let T be the inversion of point K in circle P. Point G is the midpoint of segment TE. What is the length of segment GI? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center H and circumradius 4. Let G be the reflection of point E across point D. Let J be the perpendicular bisector of segment GF. Line I is the angle bisector of angle HCF. Let W be the reflection of line J across line I. Let X be the reflection of line W across line J. Point P is the reflection of point H across line X. Find the distance between points P and D. 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 K be the circumcircle of polygon ABCDEFGH. Let W be the midpoint of segment QC. Let N be the perpendicular bisector of segment DW. Circle Y is the reflection of circle K across line N. Let V be the inversion of point G in circle Y. What is the distance between points W and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDFG is a regular hexagon with center E and circumradius 10. Line P passes through point C parallel to segment AB. Let Q be the midpoint of segment AB. Point J is the reflection of point Q across line P. What is the distance between points D and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJLM be a regular dodecagon with center K and circumradius 2. Let V be the rotation of point L by angle pi/6 around point E. Point T is the reflection of point I across point C. Let W be the perpendicular bisector of segment ML. Let S be the reflection of line W through point V. Point Q is the reflection of point T across line S. Find the length of segment TQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFH be a regular 7-gon with center G and circumradius 8. Point K is the midpoint of segment AD. Point L is the reflection of point F across point H. Let V be the line through points L and K. Let Q be the reflection of line V through point E. Point N is the reflection of point D across line Q. What is the length of segment CN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center R and circumradius 3. Let T be the vector from point E to point J. Point Z is the translation of point A by vector T. Let V be the midpoint of segment IE. Line S passes through points J and A. Let Q be the circumcircle of polygon ABCDEFGHIJ. Point Y is the reflection of point V across line S. Point W is the inversion of point Y with respect to circle Q. Find the distance between points W and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 4. Point Y is the midpoint of segment IU. Segments CG and EI intersect at point X. What is the distance between points X and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHI be a regular 9-gon with center T and circumradius 5. Let V be the line through points B and T. Let J be the rotation of point E by angle pi/3 around point G. Let O be the midpoint of segment DJ. Line L passes through point E perpendicular to line V. Point U is the reflection of point O across line L. Point Q is the reflection of point I across segment UO. Find the distance between points B and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center U and circumradius 5. Let S be the line through points K and L. Point O is the reflection of point E across line S. Point Z is the reflection of point F across segment OB. What is the length of segment HZ? 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. Let X be the angle bisector of angle LEH. Point V is the reflection of point L across line X. Line U passes through points A and B. Line M is the reflection of line U across point H. Point Q is the reflection of point K across line M. 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 ABCDEFGHI be a regular 9-gon with center X and circumradius 7. Line W is the angle bisector of angle DHE. Point K is the reflection of point E across line W. Let S be the midpoint of segment KG. Point R is the reflection of point H across point S. What is the length of segment HR? 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. Point Z is the midpoint of segment GD. Let W be the reflection of point C across point G. Line O is the angle bisector of angle ZWH. Let J be the midpoint of segment CB. Point Q is the midpoint of segment HJ. Point N is the reflection of point B across line O. Point I is the midpoint of segment QG. Point X is the midpoint of segment NI. 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]
ABCDEFGHI is a regular 9-gon with center W and circumradius 9. Line V is the angle bisector of angle IGE. Let T be the line through point A perpendicular to line V. Point X is the reflection of point F across line T. Point U is the reflection of point X across line V. Find the length of segment DU. 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 1. Let P be the midpoint of segment AC. Let U be the midpoint of segment DB. Let T be the midpoint of segment UP. What is the length of segment ET? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ACDEF is a regular 5-gon with center B and circumradius 12. Let Y be the reflection of point F across point C. Let V be the perpendicular bisector of segment BA. Line G is the reflection of line V across point Y. Point Q is the reflection of point C across line G. Find the length of segment QF. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center W and circumradius 8. Point Q is the midpoint of segment HC. Point S is the reflection of point D across point F. Let K be the circumcircle of polygon ABCDEFGH. Point N is the inversion of point S with respect to circle K. Point Z is the midpoint of segment BN. Find the distance between points Z and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center L and circumradius 6. Let O be the circumcircle of polygon ABCDEFGH. Let S be the angle bisector of angle FAB. Circle P is the reflection of circle O across line S. Let Z be the inversion of point B in circle P. What is the distance between points E 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 2. Let R be the circumcircle of polygon ABCDEFGIJKLM. Circle Q is the reflection of circle R across point B. Let Y be the line through point M perpendicular to segment BI. Let P be the reflection of circle Q across line Y. Let T be the inversion of point G in circle P. Find the measure of angle TIM, 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 ABCDEFHIJ be a regular 9-gon with center G and circumradius 1. Line Y passes through point A parallel to segment EF. Point L is the reflection of point D across line Y. Let Z be the reflection of point G across point L. Line R is the angle bisector of angle GFI. Point P is the reflection of point B across line R. Find the distance between points P and Z. 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 9. Let E be the perpendicular bisector of segment CY. Let Z be the reflection of line E through point C. Let T be the line through points C and B. Point S is the intersection of lines Z and T. Find the distance between points D and S. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center U and circumradius 6. Let J be the circumcircle of polygon ABCDEFG. Line R is the angle bisector of angle CGE. Line L passes through point E parallel to line R. Line O is the reflection of line R across line L. Let Q be the reflection of circle J across line O. Let I be the inversion of point A in circle Q. What is the distance between points I and B? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFGH is a regular 7-gon with center A and circumradius 12. Let Y be the midpoint of segment GC. Line L passes through point Y parallel to segment ED. Point I is the reflection of point B across line L. Point X is the reflection of point I across segment ED. Point U is the reflection of point X across point Y. What is the length of segment UD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEFG is a regular hexagon with center A and circumradius 4. Let Q be the line through point G parallel to segment CE. Point V is the reflection of point B across segment CE. Line K passes through points E and V. Lines Q and K intersect at point R. Find the distance between points R and E. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 6. Let P be the reflection of point I across point H. Let Q be the midpoint of segment MP. Point Z is the midpoint of segment IB. Find the distance between points Q 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 ABCDEFGHIJK be a regular 11-gon with center T and circumradius 4. Line O is the angle bisector of angle DIE. Point V is the reflection of point G across line O. Let N be the midpoint of segment GB. What is the distance between points N and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center L and circumradius 11. Let P be the line through points A and B. Line Y passes through point F parallel to line P. Let W be the reflection of line Y across line P. Point K is the reflection of point A across line W. What is the distance between points L 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 Q and circumradius 6. Let M be the reflection of point D across point E. Let Z be the midpoint of segment QC. Let G be the angle bisector of angle ZAQ. Point U is the reflection of point Z across line G. What is the distance between points U and M? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center G and circumradius 6. Let X be the circumcircle of polygon ABCDEF. Point I is the midpoint of segment FE. Let H be the inversion of point I in circle X. What is the length of segment CH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFG be a regular 7-gon with center M and circumradius 4. Point W is the reflection of point C across point M. Let T be the line through point M perpendicular to segment AF. Point V is the reflection of point W across line T. Find the measure of angle EVB, in radians. 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 1. Let O be the circumcircle of polygon ABCDEFGHI. Point V is the reflection of point E across segment HB. Point U is the inversion of point V with respect to circle O. 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 ABCDE be a regular 5-gon with center Q and circumradius 5. Point X is the reflection of point Q across point D. Line V passes through points X and Q. Point L is the midpoint of segment AX. Point N is the reflection of point L across line V. Find the length of segment XN. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center R and circumradius 8. Line W is the angle bisector of angle ECF. Point O is the midpoint of segment RD. Let J be the perpendicular bisector of segment RD. Point V lies on line J. Let M be the line through point O perpendicular to line W. Point P is the reflection of point C across line M. Find the measure of angle VCP, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center U and circumradius 1. Line T passes through point C perpendicular to segment CA. Point L is the reflection of point D across line T. Line H is the perpendicular bisector of segment LC. Point R is the reflection of point B across line H. What is the distance between points R 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 ABCDF be a regular 5-gon with center E and circumradius 10. Let I be the midpoint of segment BC. Point J is the midpoint of segment IF. Let M be the circumcircle of polygon ABCDF. Point T is the inversion of point J with respect to circle M. Find the distance between points T 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 BCDEF be a regular 5-gon with center A and circumradius 9. Let R be the perpendicular bisector of segment FA. Point Q is obtained by rotating point D by angle 11pi/12 about point E. Let N be the line through points D and B. Line J is the reflection of line R across point Q. Line T is the reflection of line J across point F. Point V is the intersection of lines T and N. Let S be the midpoint of segment AD. Find the length of segment VS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center J and circumradius 12. Line P is the angle bisector of angle JFE. Let U be the line through point B parallel to line P. Let T be the angle bisector of angle CAE. Line Y is the reflection of line T across line U. Point I is on line P. Line V is the reflection of line Y across point A. Point S is the reflection of point I across line V. Find the measure of angle ABS, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 3. Let S be the midpoint of segment LB. Let U be the line through point L perpendicular to segment LB. V is constructed so that KV = 6. Point O is on line U with |OV| = 3. Let W be the circumcircle of polygon ABCEFGHIJKLM. Let P be the reflection of line U through point S. Point Y is the inversion of point S with respect to circle W. Line N passes through point Y perpendicular to line P. Let Z be the line through point O parallel to line N. Let R be the reflection of line P across line U. Let T be the line through point H parallel to line Z. Point X is the intersection of lines T and R. Find the distance between points X and K. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCD is a square with center H and circumradius 10. Let Q be the vector from point C to point A. Point O is the translation of point D by vector Q. Let T be the perpendicular bisector of segment HB. Let V be the angle bisector of angle DCO. Let I be the reflection of line V through point D. Let Y be the reflection of line T across line I. Point U is the midpoint of segment BA. Point P is the midpoint of segment HB. Point J is the reflection of point P across segment BA. Point M is the reflection of point B across line Y. Find the measure of angle MJU, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 7. Line S passes through points I and H. Let X be the perpendicular bisector of segment DJ. Line P is the reflection of line S across line X. Line O passes through points B and J. Lines P and O intersect at point M. Find the distance between points H and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJ be a regular 10-gon with center X and circumradius 3. Point V is the reflection of point G across point E. Line Z passes through points D and V. Line Z and segment AX intersect at point M. What is the distance between points J and M? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFGHI be a regular octagon with center B and circumradius 11. Let L be the reflection of point A across point C. Point V is the midpoint of segment DI. Point T is the midpoint of segment VD. What is the length of segment LT? 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 6. Point N is the midpoint of segment FE. Let J be the midpoint of segment BN. What is the length of segment FJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BCDEFGHIJKLM be a regular dodecagon with center A and circumradius 9. Let R be the midpoint of segment LI. Line U is the perpendicular bisector of segment IM. Let Q be the line through point J parallel to line U. Point V is the reflection of point D across line Q. What is the distance between points V and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCEFGHI is a regular octagon with center D and circumradius 3. Let T be the perpendicular bisector of segment DG. Let W be the line through point I perpendicular to line T. Point J is the reflection of point E across point H. Point U is the reflection of point H across line W. Let L be the midpoint of segment CJ. Let O be the angle bisector of angle LUF. Line R passes through point F parallel to line O. Point K is the intersection of line R and segment IG. What is the distance between points C and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center R and circumradius 12. Let M be the reflection of point E across point R. Let W be the perpendicular bisector of segment MF. Line J passes through point D parallel to line W. Point P is the reflection of point F across line J. Let I be the midpoint of segment DM. Find the distance between points I 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 I and circumradius 2. Let W be the midpoint of segment GA. Point V is the midpoint of segment HA. Let T be the perpendicular bisector of segment VD. Point X is the reflection of point G across line T. Point S is the reflection of point I across segment BW. 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]
ABCDEFHI is a regular octagon with center G and circumradius 6. Let R be the circumcircle of polygon ABCDEFHI. Let T be the reflection of circle R through point E. Let V be the inversion of point A in circle T. Point P is the midpoint of segment VE. What is the measure of angle PAB, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center X and circumradius 6. Line Z is the angle bisector of angle ABD. Let V be the line through point D parallel to line Z. Let K be the line through point A parallel to line V. Point R is the reflection of point C across line K. What is the distance between points C 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 ABDEFGHIJKL be a regular 11-gon with center C and circumradius 9. Let Q be the midpoint of segment FA. Point W is the reflection of point Q across segment IH. Point R is the reflection of point C across point F. Find the distance between points R and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFGHI be a regular octagon with center C and circumradius 6. Line S is the perpendicular bisector of segment DF. Point W is the midpoint of segment EB. Point R is the reflection of point W across line S. Find the length of segment RA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center Y and circumradius 10. Let O be the circumcircle of polygon ABCDEFGH. Line U passes through points A and C. Point J is the reflection of point D across line U. Point T is the inversion of point J with respect to circle O. Line R passes through point D parallel to segment EC. Point M is the reflection of point C across line R. Find the length of segment TM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center R and circumradius 12. Let M be the circumcircle of polygon ABCDEFGHIJKL. Circle O is the reflection of circle M across point A. Line P passes through points H and K. Point X is the reflection of point F across line P. Let Y be the inversion of point X in circle O. What is the length of segment YR? Express your answer as a decimal number rounded to 2 decimal places.

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