[User]
ABCDEFGH is a regular octagon with center I and circumradius 4. Let L be the midpoint of segment GD. Let W be the midpoint of segment LI. Line P passes through points W and B. Line P and segment GE intersect at point M. Find the distance between points B 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 U and circumradius 4. Let M be the midpoint of segment DB. Line Q is the angle bisector of angle JIG. Point N is the reflection of point M across line Q. Point O is the midpoint of segment NJ. Find the distance between points U and O. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJK is a regular 11-gon with center X and circumradius 5. Line N passes through point X perpendicular to segment XK. Point U is the reflection of point A across line N. Line O passes through point U perpendicular to segment XK. Let W be the reflection of point G across point F. Point Q is the reflection of point W across line O. What is 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]
ABCDEFGHIJKL is a regular dodecagon with center N and circumradius 10. Point S is the reflection of point K across point E. Let R be the angle bisector of angle AJK. Point X is the reflection of point S across line R. Point Q is the reflection of point X across point H. Find the length of segment QB. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center L and circumradius 12. Line S is the perpendicular bisector of segment LG. Line K is the reflection of line S across point H. Point V is the reflection of point F across line K. What is 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 ABCDE be a regular 5-gon with center I and circumradius 3. Let N be the perpendicular bisector of segment IB. Point M is the reflection of point C across line N. Point L is the midpoint of segment IC. Point U is the reflection of point M across point L. What is the length of segment EU? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center T and circumradius 6. Let Y be the reflection of point D across point A. Line Z is the angle bisector of angle HFY. Point O is the midpoint of segment CK. Point P is the reflection of point O across line Z. What is the length of segment OP? 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 M and circumradius 5. Line W is the angle bisector of angle GAM. Let Z be the rotation of point I by angle pi/12 around point D. Point V lies on line W at distance 4 from point F. Line L is the angle bisector of angle JEZ. Let P be the reflection of line L across line W. Line R is the perpendicular bisector of segment GB. Let T be the line through point V perpendicular to line R. Line S passes through point C parallel to line T. Lines S and P intersect at point Y. Find the distance between points Y and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center J and circumradius 1. Let V be the midpoint of segment CD. Let M be the midpoint of segment AC. Find the length of segment MV. 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 3. Let M be the rotation of point A by angle pi/12 around point F. Let P be the line through points A and M. Let T be the circumcircle of polygon ABCDEF. Circle U is the reflection of circle T across line P. Point I is the inversion of point C with respect to circle U. Point Y is the reflection of point I across point M. Find the length of segment JY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center N and circumradius 8. Let V be the circumcircle of polygon ABCDEFGHIJKL. Line Y passes through points I and A. Point Z is the midpoint of segment NC. Let W be the perpendicular bisector of segment DC. Let U be the reflection of line W across line Y. Let R be the inversion of point Z in circle V. Point P is the reflection of point D across line U. What is the measure of angle RJP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDF is a regular 5-gon with center E and circumradius 4. Let N be the midpoint of segment BC. Let W be the perpendicular bisector of segment EF. Point M is the reflection of point A across line W. Point I is the reflection of point A across point M. What is the length of segment NI? 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 W and circumradius 6. Let H be the line through points B and A. Point X is the reflection of point A across point E. Line T is the reflection of line H across point W. Line M is the reflection of line H across line T. Point Y is the reflection of point X across point A. Point J is the reflection of point Y across line M. Point P is the reflection of point B across point J. Find the distance between points P and W. 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 V and circumradius 5. Let W be the perpendicular bisector of segment BE. Line P passes through point C perpendicular to line W. Let U be the reflection of point F across point A. Point M is the reflection of point K across line P. 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]
ABCDEFGIJ is a regular 9-gon with center H and circumradius 8. Let R be the perpendicular bisector of segment AI. Line P is the reflection of line R across point B. Let Q be the line through point A perpendicular to line P. Point O is the reflection of point E across line Q. What is the distance between points O and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center S and circumradius 3. Point P is the midpoint of segment SB. Let F be the reflection of point C across point S. Let W be the angle bisector of angle APE. Point K is the reflection of point F across line W. Find the length of segment PK. 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 9. Point W is the reflection of point E across point C. Let L be the angle bisector of angle WFX. Line S passes through point B parallel to line L. Let T be the reflection of line S across line L. Line Y is the reflection of line T across point C. Point O is the reflection of point B across line Y. What is the distance between points O and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFGHI be a regular octagon with center C and circumradius 6. Point Y is the midpoint of segment IG. Point O is the reflection of point Y across segment BD. Point R is the midpoint of segment OY. What is the distance between points A and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center J and circumradius 3. Line V passes through point J parallel to segment CE. Point H is the midpoint of segment AE. Line T passes through points D and H. Let O be the circumcircle of polygon ABCDE. Let Q be the line through points D and C. Circle X is the reflection of circle O across line T. Point M is the reflection of point H across segment CE. Let K be the line through point E parallel to line Q. Lines K and V intersect at point S. Circle W is the reflection of circle X across point S. Point I is the reflection of point C across point M. Let R be the inversion of point A in circle W. What is the distance between points I and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center V and circumradius 11. Let Q be the circumcircle of polygon ABCDEFGHIJKL. Let P be the midpoint of segment GF. Let X be the midpoint of segment GI. Point R is the inversion of point P with respect to circle Q. What is the distance between points R and X? 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 11. Let R be the line through points A and G. Line Q passes through points C and B. Line K is the reflection of line R across line Q. Point P is the reflection of point D across line K. What is the measure of angle PBE, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFHIJKL is a regular 11-gon with center G and circumradius 1. Let N be the midpoint of segment KL. Let X be the midpoint of segment EC. Let P be the line through point K perpendicular to segment NX. Let Z be the midpoint of segment NX. Let W be the reflection of point L across point C. Point O is the reflection of point D across line P. Let R be the reflection of point W across point O. Find 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 ABCDEFGH be a regular octagon with center M and circumradius 3. Line Y is the angle bisector of angle BEM. Let R be the line through point M parallel to line Y. Line O passes through points A and F. Lines R and O intersect at point Q. What is the distance between points Q 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 ABCDEFGHIJK be a regular 11-gon with center V and circumradius 1. Point L is the reflection of point F across point J. Line X passes through points A and K. Let N be the line through point I parallel to line X. Point M is the reflection of point L across line N. 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]
ABCDEFGHI is a regular 9-gon with center M and circumradius 9. Let V be the line through points F and E. Let P be the angle bisector of angle AFG. Let Z be the perpendicular bisector of segment CI. Point N is the reflection of point I across line V. Let Q be the reflection of point C across point F. Let T be the reflection of line P across line Z. Point Y is the reflection of point N across line T. What is the length of segment QY? 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 S and circumradius 10. Point T is the midpoint of segment BF. Point W is the reflection of point S across point C. Let X be the midpoint of segment DT. Find 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]
ABCDEGHI is a regular octagon with center F and circumradius 2. Let L be the reflection of point G across point I. Let V be the line through points L and H. Point K is the reflection of point D across line V. Point U is the reflection of point G across line V. Let T be the midpoint of segment AU. Point P is the reflection of point K across point H. What is the distance between points T and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFGHIJKLM is a regular dodecagon with center C and circumradius 8. Let R be the circumcircle of polygon ABDEFGHIJKLM. Let P be the perpendicular bisector of segment AC. Circle X is the reflection of circle R across line P. Let W be the line through point I perpendicular to line P. Point O is the inversion of point G with respect to circle X. Line N passes through point O parallel to line W. Segment BK and line N intersect at point V. What is the length of segment FV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEF be a regular 5-gon with center C and circumradius 2. Point L is the reflection of point A across point E. Let H be the midpoint of segment LB. Point X is the midpoint of segment HD. What is the distance between points X and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center Q and circumradius 10. Point H is the reflection of point Q across point E. Let Z be the angle bisector of angle ECA. Point G is the reflection of point H across line Z. Point I is the reflection of point C across point G. Find the length of segment DI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center K and circumradius 2. Point R is the reflection of point E across point A. Point W is the reflection of point K across point A. Line J is the perpendicular bisector of segment WR. Point I is the reflection of point C across line J. Find the length of segment WI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center N and circumradius 8. Let O be the line through points B and G. Let M be the midpoint of segment AD. Let L be the midpoint of segment DM. Point X is the reflection of point N across line O. Point P is the midpoint of segment NX. Find the distance between points P and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center M and circumradius 12. Line U is the angle bisector of angle DCB. Let O be the reflection of line U through point B. Point F is the reflection of point A across point D. Let Z be the reflection of point F across point A. Let R be the line through point A perpendicular to line O. Point W is the reflection of point B across line R. Point G is the midpoint of segment DZ. Find 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]
ABCDEFGHIJKL is a regular dodecagon with center O and circumradius 5. Line V passes through points F and O. Line R passes through points G and J. Let T be the reflection of line R across line V. Point M is the reflection of point I across line T. What is the measure of angle FDM, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center K and circumradius 4. Let S be the perpendicular bisector of segment KA. Line J passes through point F perpendicular to line S. Point T is the reflection of point K across line J. Let Q be the midpoint of segment TE. 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 ABCDE be a regular 5-gon with center O and circumradius 9. Let K be the angle bisector of angle CEB. Let Q be the line through point C perpendicular to line K. Point G is the reflection of point D across line Q. Point U is the midpoint of segment AE. Point V is the midpoint of segment EG. What is the distance between points V and U? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center M and circumradius 10. Line Z passes through points E and M. Let V be the rotation of point C by angle 11pi/12 around point D. Line P passes through point H parallel to line Z. Line S is the reflection of line Z across line P. Point R is the reflection of point F across line S. Point O is the midpoint of segment VR. What is the distance between points V and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJ be a regular 10-gon with center S and circumradius 11. Let V be the midpoint of segment EB. Point O is the midpoint of segment EV. Point X is the reflection of point F across segment SO. What is the length of segment XB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDE be a square with center B and circumradius 4. Let G be the reflection of point B across point E. Line U is the perpendicular bisector of segment EB. Line Z is the reflection of line U across point G. Point X is the reflection of point D across line Z. Point T is the reflection of point X across segment AC. What is the length of segment XT? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 6. Let Y be the circumcircle of polygon ABCDEFGIJKLM. Line S passes through points K and A. Let N be the reflection of circle Y through point M. Let V be the line through points G and M. Lines S and V intersect at point W. Let X be the inversion of point J in circle N. What is the distance between points X and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center P and circumradius 10. Line Q is the perpendicular bisector of segment GI. Line M passes through point F parallel to line Q. Point T is the reflection of point G across point F. Point V is the reflection of point T across line M. What is 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 ABCDEFGH be a regular octagon with center Q and circumradius 2. Let M be the rotation of point Q by angle 3pi/4 around point B. Let P be the angle bisector of angle GMD. Let V be the line through points F and E. Let K be the reflection of line P through point H. Line U is the reflection of line P across line K. Lines U and V intersect at point X. What is the length of segment DX? 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 I and circumradius 8. Line U passes through points G and C. Let W be the line through point A perpendicular to line U. Point Y is the reflection of point B across line W. Point H is the reflection of point Y across line U. Let O be the midpoint of segment GC. What is the distance between points O 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 ABCE be a square with center D and circumradius 7. Point K is the reflection of point E across point B. Let U be the circumcircle of polygon ABCE. Let S be the inversion of point K in circle U. Point V is the midpoint of segment KD. Find the distance between points S 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 U and circumradius 9. Line K passes through points C and A. Line Z passes through point E parallel to line K. Line N passes through point U parallel to line K. Let M be the reflection of line N across line Z. Point I is the reflection of point F across line M. Find the distance between points I and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDFGHIJKLM be a regular dodecagon with center E and circumradius 6. Let Z be the line through points C and F. Let N be the reflection of line Z through point A. Point S is the intersection of segment EK and line N. What is the length of segment AS? 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 8. Let N be the reflection of point Q across point H. Let M be the reflection of point I across point E. Point U is the midpoint of segment NH. 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 ABCDEFGHI be a regular 9-gon with center K and circumradius 8. Line W passes through points I and D. Point J is the intersection of segment AG and line W. Let V be the midpoint of segment AG. What is the measure of angle IJV, in radians? 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 3. Line O is the angle bisector of angle FHA. Line X passes through point K parallel to line O. Line X and segment LB intersect at point M. Find the length of segment UM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFH is a regular 7-gon with center G and circumradius 3. Let U be the circumcircle of polygon ABCDEFH. Let Y be the reflection of circle U through point A. Let J be the inversion of point H in circle Y. Let I be the reflection of point J across point H. Find the length of segment EI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ET = 1. ABCDFG is a regular hexagon with center T and circumradius 3. Let Q be the perpendicular bisector of segment BT. Point Z is the intersection of segments DG and FB. Point W is the reflection of point Z across line Q. What is the length of segment WC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center L and circumradius 9. Let Z be the circumcircle of polygon ABCDEF. Let Y be the line through points B and L. Circle K is the reflection of circle Z across point F. Let U be the reflection of circle K across line Y. Point Q is the inversion of point B with respect to circle U. Let N be the inversion of point Q in circle K. What is the length of segment QN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHI is a regular 9-gon with center X and circumradius 5. Line U is the angle bisector of angle FIA. Line Z passes through point X parallel to line U. Let O be the reflection of line Z across line U. Line O and segment EA intersect at point V. What is the measure of angle HAV, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center U and circumradius 5. Point K is obtained by rotating point B by angle pi/3 about point A. Line S passes through points E and G. Let I be the perpendicular bisector of segment KH. Line Q is the reflection of line S across point A. Let Y be the reflection of line Q across line I. Point T is the reflection of point A across line Y. Point J is the reflection of point T across point U. Point V is the midpoint of segment AD. What is the length of segment JV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJK be a regular 11-gon with center Z and circumradius 2. Let Y be the line through points I and A. Line N is the reflection of line Y across point H. Point L is the reflection of point A across line N. Let P be the angle bisector of angle FEJ. Let R be the midpoint of segment LK. Point U is the reflection of point R across line P. What is the distance between points U 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 ABCD be a square with center O and circumradius 3. Point L is the midpoint of segment DC. Let W be the circumcircle of polygon ABCD. Circle E is the reflection of circle W across point D. Let J be the inversion of point O in circle E. Point I is the reflection of point B across point L. Find the distance between points I 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 ABCDEFGHIKL be a regular 11-gon with center J and circumradius 10. Let R be the midpoint of segment LG. Point O is the midpoint of segment RL. Let P be the midpoint of segment GO. Find the distance between points P 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 ABCDEFGH be a regular octagon with center U and circumradius 3. Let J be the midpoint of segment CA. Let W be the midpoint of segment AU. Line X is the angle bisector of angle JCW. Point M is the reflection of point D across line X. Point Z is the midpoint of segment FM. Find the length of segment AZ. 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 1. Let L be the circumcircle of polygon ABCDEF. Line Y is the angle bisector of angle AFC. Let G be the midpoint of segment BD. Let M be the reflection of circle L through point F. Point T is the midpoint of segment CG. Circle X is the reflection of circle M across line Y. Let I be the inversion of point T in circle X. Find the distance between points B and I. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ACDEFG be a regular hexagon with center B and circumradius 8. Let K be the perpendicular bisector of segment FE. Let Z be the line through point D parallel to line K. Let R be the reflection of line Z across line K. Point Q is the reflection of point D across line R. What is the distance between points Q 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 M and circumradius 7. Line T is the angle bisector of angle DGF. Point J is the reflection of point D across line T. Line X is the perpendicular bisector of segment JF. Point N is the reflection of point D across point B. Line L passes through point H perpendicular to line X. Point Z is the reflection of point N across line L. Find the distance between points G 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 11. Let V be the circumcircle of polygon ABCDFGHI. Line P passes through points I and C. Circle J is the reflection of circle V across line P. Point U is the inversion of point B with respect to circle J. What is the distance between points U and C? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 4. Let O be the line through point K parallel to segment BE. Let R be the midpoint of segment BE. Point N is the reflection of point E across line O. Find the distance between points N and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEFGHI be a regular octagon with center D and circumradius 10. Let W be the circumcircle of polygon ABCEFGHI. Let Z be the angle bisector of angle FED. Circle M is the reflection of circle W across point I. Point R is the reflection of point D across point C. Let Y be the inversion of point R in circle M. Let V be the reflection of line Z through point A. Point T is the reflection of point Y across line V. What is 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]
ABCDEFGH is a regular octagon with center O and circumradius 10. Let S be the reflection of point G across point A. Let N be the angle bisector of angle OFB. Let J be the line through points A and B. Let T be the line through point E perpendicular to line N. Let K be the reflection of line J across line T. Lines J and K intersect at point I. What is the length of segment SI? 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 11. Let Y be the midpoint of segment EF. Point R is the midpoint of segment BY. Let H be the line through points R and A. Point V is the reflection of point D across line H. What is the distance between points V and A? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center W and circumradius 3. Point T is the reflection of point A across point B. Let Q be the line through points E and W. Point X is the reflection of point T across line Q. Let U be the midpoint of segment WD. Find the distance between points U and X. 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 11. Let J be the line through points H and D. Let O be the line through point H perpendicular to line J. Point U is the midpoint of segment EA. Line Z is the reflection of line O across point B. Let M be the line through point D parallel to line Z. Point T is the reflection of point U across line M. What is the length of segment TH? 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 X and circumradius 12. Point Z is the reflection of point X across segment DA. Let U be the line through points B and C. Point O is the reflection of point Z across line U. Find the distance between points Z 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 11. Let Z be the circumcircle of polygon ABCDEFGH. Let R be the reflection of circle Z through point H. Line I passes through points X and F. Circle O is the reflection of circle R across line I. Let P be the inversion of point E in circle O. Find the distance between points P and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center X and circumradius 7. Let U be the perpendicular bisector of segment HE. Let O be the line through points J and D. Line T is the reflection of line O across line U. Let W be the line through point D perpendicular to line T. Point Z is the reflection of point J across line W. 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]
ABCDEFG is a regular 7-gon with center U and circumradius 12. Point S is the midpoint of segment BE. Let P be the circumcircle of polygon ABCDEFG. Let T be the reflection of circle P through point F. Let J be the angle bisector of angle UGS. Line K is the reflection of line J across point D. Let W be the reflection of point U across point F. Let R be the reflection of point W across point C. Circle H is the reflection of circle T across line K. Point M is the inversion of point E with respect to circle H. What is 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]
ABCDEF is a regular hexagon with center L and circumradius 6. Point V is the midpoint of segment AC. Let P be the midpoint of segment VD. Point J is the reflection of point F across segment AP. Find the length of segment BJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center T and circumradius 8. Line R passes through points A and C. Line X passes through point D parallel to line R. Let O be the perpendicular bisector of segment FA. Let U be the reflection of line O across line X. Lines X and U intersect at point W. Find the measure of angle EWG, 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 ABCDEFGHIJ be a regular 10-gon with center S and circumradius 11. Let P be the angle bisector of angle CFD. Line Y is the reflection of line P across point H. Line K passes through point A perpendicular to line Y. Line T passes through point D perpendicular to line K. Let W be the reflection of line T through point H. Point N is the reflection of point J across line W. What is the length of segment JN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHJK be a regular 10-gon with center I and circumradius 6. Point X is the midpoint of segment CG. Point P is the reflection of point X across segment BD. Point M is the midpoint of segment BD. Point U is the midpoint of segment DM. What is the distance between points U and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFG is a regular 7-gon with center M and circumradius 1. Let O be the circumcircle of polygon ABCDEFG. Circle I is the reflection of circle O across point E. Point Q is the inversion of point G with respect to circle I. Let V be the reflection of point Q across point F. What is the length of segment VE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center V and circumradius 9. Z is placed 2 units away from K. Line R is the angle bisector of angle ZKC. Point P is the midpoint of segment CK. Line X passes through point C perpendicular to line R. Point M is the midpoint of segment DP. Point S is the reflection of point M across line X. Find the distance between points C 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 I and circumradius 3. Let S be the reflection of point B across point D. Let W be the reflection of point F across point S. Line P is the angle bisector of angle ACG. Point O is the reflection of point W across line P. 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 ABCDEF be a regular hexagon with center Z and circumradius 10. Point X is the reflection of point A across point E. Point N is the midpoint of segment ZD. Point H is the reflection of point X across segment NZ. Find the measure of angle NAH, 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 ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 1. Point U is the reflection of point M across point E. Line T is the angle bisector of angle KMJ. Point Y is the reflection of point K across line T. Point O is the reflection of point F across point U. What is the distance between points O and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center G and circumradius 5. Line L is the perpendicular bisector of segment CG. Line Z passes through point E perpendicular to segment DC. Let N be the midpoint of segment DC. Line Q is the reflection of line L across line Z. Point R is the reflection of point N across line Q. Find 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]
ABCDEF is a regular hexagon with center H and circumradius 4. Let R be the midpoint of segment FA. Line O is the perpendicular bisector of segment RH. Point Y is the reflection of point C across point D. Point Q is the reflection of point F across line O. Find the distance between points Y and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center Z and circumradius 7. Let I be the midpoint of segment ZF. Point Q is the midpoint of segment CI. Point H is the reflection of point D across point F. What is the length of segment HQ? 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. Line O is the angle bisector of angle GJD. Point S is the reflection of point H across line O. Let W be the circumcircle of polygon ABCDEFGHIJ. Point K is the inversion of point S with respect to circle W. Point U is the reflection of point D across point B. Find the length of segment KU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center I and circumradius 3. Let N be the reflection of point E across point D. Line J passes through points I and A. Point G is the reflection of point N across line J. Line S is the angle bisector of angle BNG. Let R be the reflection of line J across line S. Line R and segment EG intersect at point P. Find the distance between points P and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center O and circumradius 6. Line X is the angle bisector of angle ECF. Let U be the circumcircle of polygon ABCDEF. Let V be the reflection of circle U across line X. Let P be the inversion of point A in circle V. Let R be the midpoint of segment DP. Find the length of segment ER. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJK be a regular 11-gon with center V and circumradius 10. Line N passes through points A and H. Point L is the reflection of point B across line N. Point Q is the midpoint of segment HL. Find the length of segment VQ. 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 4. Point V is the reflection of point B across point H. Let W be the reflection of point H across point G. Point N is the reflection of point V across segment LA. Find the distance between points N and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center U and circumradius 6. Point P is the reflection of point B across point A. Point X is obtained by rotating point A by angle pi/6 about point G. Let W be the midpoint of segment XF. Point L is the midpoint of segment UW. Point V is the reflection of point L across segment GU. What is the distance between points V and P? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center U and circumradius 1. Let H be the midpoint of segment UB. Let M be the reflection of point C across point A. Let K be the angle bisector of angle UMC. Point R is on line K with |RU| = 8. Line J passes through point H perpendicular to line K. Point W is obtained by rotating point M by angle 5pi/6 about point C. Let L be the circumcircle of polygon ABCD. Point F is the reflection of point W across point D. Let Y be the line through point F perpendicular to line J. Let O be the reflection of line Y across line K. Let S be the reflection of circle L across line O. Let E be the inversion of point R in circle S. Find the distance between points E and H. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDF be a regular 5-gon with center E and circumradius 11. Point J is the midpoint of segment EB. Let W be the perpendicular bisector of segment CA. Let P be the line through point A perpendicular to line W. Point I is the reflection of point J across line P. 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]
ABDEFGH is a regular 7-gon with center C and circumradius 9. Let S be the perpendicular bisector of segment GC. Let M be the line through point A parallel to line S. Let Y be the reflection of line S across line M. Point O is the reflection of point A across line Y. Find the length of segment OH. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEFGHIJK be a regular 10-gon with center D and circumradius 8. Let T be the line through point G parallel to segment DK. Point V is the reflection of point B across line T. Let N be the midpoint of segment DK. Let O be the reflection of point N across point J. Find the length of segment OV. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEGHI is a regular octagon with center F and circumradius 4. Let J be the midpoint of segment HI. Let K be the circumcircle of polygon ABCDEGHI. Point W is the inversion of point J with respect to circle K. Point V is the reflection of point W across point G. Find the distance between points V 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 ABDE be a square with center C and circumradius 2. Let Z be the reflection of point B across point D. Let V be the angle bisector of angle ZBE. Line I passes through point E perpendicular to line V. Point N is the reflection of point Z across line I. What is the measure of angle ZNB, 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 ACDEFG be a regular hexagon with center B and circumradius 9. Let J be the perpendicular bisector of segment CG. Let N be the line through point F parallel to line J. Point R is the midpoint of segment BA. Point H is the reflection of point R across line N. What is the measure of angle HCB, 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 ABCDEG be a regular hexagon with center F and circumradius 8. Point T is the midpoint of segment DG. Let N be the angle bisector of angle CGD. Point P is the reflection of point F across line N. Point H is the reflection of point G across point P. What is the distance between points H 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 L and circumradius 4. Let X be the perpendicular bisector of segment EB. Let U be the line through point C parallel to line X. Line V passes through point A perpendicular to line U. Point N is the reflection of point L across line V. What is the measure of angle NAD, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center K and circumradius 11. Point Q is obtained by rotating point F by angle pi/4 about point B. Let W be the angle bisector of angle BGK. Line O passes through point C parallel to line W. Line O and segment QD intersect at point J. 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]
ZN = 12. ABCD is a square with center N and circumradius 4. Let I be the reflection of point B across point D. Let L be the reflection of point A across point I. Let T be the circumcircle of polygon ABCD. Point O is the inversion of point L with respect to circle T. What is the length of segment LO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center I and circumradius 3. Point T is the midpoint of segment HC. Segments BE and GD intersect at point Q. What is 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 ABCDEFGHI be a regular 9-gon with center U and circumradius 10. Let V be the perpendicular bisector of segment BC. Let P be the midpoint of segment FE. Point R is the reflection of point P across line V. Find the distance between points B and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center R and circumradius 9. Line F passes through points E and R. Line T is the reflection of line F across point C. Point L is the reflection of point C across line T. Point U is the reflection of point L across segment BA. Find the distance between points U and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFGHI is a regular octagon with center C and circumradius 12. Point Z is obtained by rotating point H by angle 11pi/12 about point G. Let S be the circumcircle of polygon ABDEFGHI. Let W be the perpendicular bisector of segment CB. Line K is the angle bisector of angle BEG. Let J be the reflection of line W across line K. Let R be the inversion of point Z in circle S. Point M is the reflection of point R across line J. What is the distance between points G and M? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDFGHIJKLM is a regular dodecagon with center E and circumradius 10. Line Z is the perpendicular bisector of segment GI. Let X be the perpendicular bisector of segment FL. Let O be the line through point I parallel to line Z. Line P is the reflection of line X across line O. Let U be the reflection of line O through point J. Point S is the intersection of lines U and P. Find the length of segment JS. 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 6. Let X be the midpoint of segment EC. Line Q is the angle bisector of angle FXC. Point V is the reflection of point E across line Q. Let S be the midpoint of segment VP. Find the distance between points S and D. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEG is a regular hexagon with center F and circumradius 7. Line M passes through points G and A. Let Z be the circumcircle of polygon ABCDEG. Circle O is the reflection of circle Z across line M. Let S be the inversion of point D in circle O. What is the distance between points C 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 O and circumradius 8. Line M passes through points O and D. Line K is the reflection of line M across point C. Let Z be the line through point C perpendicular to line K. Point U is on line Z. What is the measure of angle UEB, in radians? 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 9. Let P be the vector from point I to point D. Point Z is the translation of point F by vector P. Point L is obtained by rotating point H by angle 2pi/3 about point Z. Point Y is the translation of point A by vector P. Line M passes through points A and Y. Line X is the perpendicular bisector of segment CZ. Line R is the perpendicular bisector of segment LC. Let U be the reflection of line R across line X. Point Q is the intersection of lines U and M. Find the distance between points Q and D. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BCDEF is a regular 5-gon with center A and circumradius 12. Let Z be the rotation of point C by angle 3pi/4 around point F. Line S is the perpendicular bisector of segment ED. Let R be the line through point Z perpendicular to segment BC. Let G be the reflection of line R across line S. Point W is the reflection of point F across point C. Point N is the reflection of point W across line 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 N and circumradius 12. Let Z be the line through points E and D. Let K be the midpoint of segment NA. Let Y be the reflection of point E across point A. Let G be the reflection of line Z through point K. Line V passes through points C and Y. Let I be the line through point A perpendicular to line V. Let F be the line through point N parallel to line I. Lines F and G intersect at point T. Find the distance between points K 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 ABCDFGHIJ be a regular 9-gon with center E and circumradius 6. Let P be the angle bisector of angle GAC. Let R be the circumcircle of polygon ABCDFGHIJ. Circle K is the reflection of circle R across line P. Let Z be the inversion of point C in circle K. Let N be the inversion of point Z in circle R. What is the length of segment NH? 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 8. Point V is the reflection of point E across point A. Point R is the midpoint of segment CD. Let M be the line through points R and V. Point O is the reflection of point B across line M. What is the distance between points V and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center W and circumradius 1. Let V be the line through points E and L. Line R passes through point G perpendicular to line V. Let Q be the midpoint of segment GF. Line Y is the reflection of line R across point J. Point N is the reflection of point J across line Y. Find the length of segment NQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center W and circumradius 7. Point M is the midpoint of segment JG. Point P is the reflection of point E across point H. Point R is the reflection of point B across segment DP. What is the length of segment MR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center X and circumradius 7. Line W is the angle bisector of angle JGI. Line N is the reflection of line W across point C. Let Y be the reflection of point K across point I. Point S is the reflection of point A across line N. Point M is the reflection of point I across point S. 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]
ABCDEFGHIJK is a regular 11-gon with center U and circumradius 11. Line Q is the angle bisector of angle IGU. Point R is the reflection of point U across line Q. Let O be the circumcircle of polygon ABCDEFGHIJK. Circle S is the reflection of circle O across point H. Let Z be the inversion of point A in circle S. Find 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 ABCDEFGH be a regular octagon with center P and circumradius 10. Let J be the circumcircle of polygon ABCDEFGH. Let K be the line through points C and D. Let R be the angle bisector of angle CHF. Let Z be the line through points F and H. Lines K and Z intersect at point M. Circle O is the reflection of circle J across line R. Let Q be the inversion of point F in circle O. 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]
ABCDEFGH is a regular octagon with center J and circumradius 3. Point Z is the reflection of point G across point E. Line I is the perpendicular bisector of segment JG. Line S passes through point Z parallel to line I. Point T is the reflection of point D across line S. What is the distance between points T 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 4. Line P passes through points A and D. Let Z be the line through point F parallel to line P. Point L is the reflection of point G across line Z. Let W be the midpoint of segment KS. Find the length of segment WL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center X and circumradius 7. Let Z be the angle bisector of angle BAE. Let L be the rotation of point E by angle 3pi/4 around point B. Line T is the reflection of line Z across point L. Point W is the reflection of point B across line T. Point O is the reflection of point D across point X. Point K is the reflection of point B across point W. What is the distance between points O 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 ABCDEG be a regular hexagon with center F and circumradius 2. Let P be the circumcircle of polygon ABCDEG. Let V be the reflection of circle P through point C. Point O is the inversion of point A with respect to circle V. Let I be the midpoint of segment DO. Find the distance between points I and G. 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 9. Let R be the reflection of point K across point J. Let T be the angle bisector of angle AJR. Point P is the reflection of point B across line T. Let W be the line through points D and H. Line S passes through point F parallel to line W. Let U be the reflection of point K across point P. Point X is the intersection of line S and segment UD. What is the length of segment RX? 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 7. Point M is obtained by rotating point E by angle 5pi/12 about point B. Line W is the angle bisector of angle ICM. Point P is the reflection of point M across line W. Point K is the midpoint of segment EB. Point F is the midpoint of segment MP. What is the length of segment KF? 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 S and circumradius 11. Line M passes through points J and C. Let L be the line through point I parallel to line M. Let W be the reflection of line L across line M. Point U is the reflection of point J across line W. What is the length of segment IU? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center Q and circumradius 1. Point P is obtained by rotating point A by angle pi about point E. Point I is the reflection of point E across segment BP. Line X passes through point H perpendicular to segment BP. Point V is the reflection of point I across line X. What is 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]
ABCDEFGHI is a regular 9-gon with center Y and circumradius 12. Let J be the circumcircle of polygon ABCDEFGHI. Let S be the reflection of point E across point F. Let V be the reflection of circle J through point S. Point R is the inversion of point I with respect to circle V. Let N be the inversion of point H in circle V. What is the distance between points N and R? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDF be a regular 5-gon with center E and circumradius 1. Line N passes through points D and F. Let R be the line through point A parallel to line N. Let Q be the perpendicular bisector of segment CB. Let H be the reflection of line R across line Q. Let J be the circumcircle of polygon ABCDF. Let T be the reflection of line N across line H. Let S be the reflection of point F across point E. Let V be the reflection of circle J through point S. Point L is the reflection of point S across line T. Point M is the inversion of point L with respect to circle V. Find the length of segment MA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDE is a regular 5-gon with center W and circumradius 5. Let Y be the angle bisector of angle CED. Let P be the line through points B and E. Point S is the reflection of point D across line P. Line I is the reflection of line Y across point C. Point R is the reflection of point B across line I. Let M be the midpoint of segment ES. Find the length of segment MR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHI be a regular 9-gon with center M and circumradius 9. Point Q is the reflection of point F across point M. Let L be the perpendicular bisector of segment QI. Let S be the reflection of point Q across point I. Let N be the midpoint of segment AF. Point Z is the reflection of point N across line L. What is the length of segment SZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFHI be a regular octagon with center G and circumradius 6. Let M be the reflection of point A across point E. Let Y be the line through point M perpendicular to segment AI. Point T is the reflection of point C across line Y. Point L is the reflection of point F across segment DH. Find the distance between points L and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center V and circumradius 9. Let K be the line through points E and H. Let P be the line through point D parallel to segment BF. Lines K and P intersect at point R. What is the length of segment RV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHI be a regular 9-gon with center X and circumradius 1. Let O be the circumcircle of polygon ABCDEFGHI. Point Q is the reflection of point D across point H. Point P is the reflection of point I across point Q. Let M be the reflection of circle O through point Q. Point K is the inversion of point F with respect to circle M. 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]
Let ABCDEFGHIJ be a regular 10-gon with center U and circumradius 7. Point K is the midpoint of segment FE. Point Q is the reflection of point K across point B. Point Z is the midpoint of segment JQ. Let M be the reflection of point Z across point E. What is the length of segment BM? 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 3. Point Y is the midpoint of segment AB. Let R be the circumcircle of polygon ABCD. Let N be the inversion of point Y in circle R. Let Q be the midpoint of segment AN. What is the distance between points Q and A? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCD be a square with center E and circumradius 10. Let V be the circumcircle of polygon ABCD. Point K is obtained by rotating point E by angle 3pi/4 about point D. Line L passes through points E and K. Point I is the reflection of point C across line L. Point H is the midpoint of segment AK. Let W be the midpoint of segment IH. Let Z be the inversion of point W in circle V. 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]
ABCDEFGH is a regular octagon with center S and circumradius 2. Let V be the perpendicular bisector of segment SE. Point Z is the reflection of point G across line V. Let X be the line through points B and F. Line O passes through points H and C. Point W is the intersection of lines X and O. Line M is the reflection of line V across point W. Point P is the reflection of point F across line M. What is the length of segment ZP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center T and circumradius 4. Point U is obtained by rotating point C by angle 7pi/12 about point G. Point P is obtained by rotating point H by angle 11pi/12 about point T. Let V be the line through points U and B. Point X is the reflection of point H across line V. Let S be the reflection of point A across point X. Point W is the reflection of point C across point S. Point Y is the midpoint of segment WP. What is 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]
ABCDEFGHIJKL is a regular dodecagon with center Z and circumradius 3. Line V passes through points Z and G. Line M passes through point L parallel to line V. Let W be the reflection of line M across line V. Point X is the reflection of point A across line W. Let T be the midpoint of segment GX. What is 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 ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 3. Let Q be the perpendicular bisector of segment GL. Let R be the line through point H parallel to line Q. Point S is the reflection of point D across line R. Point W is the reflection of point S across line Q. Find the length of segment WK. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center R and circumradius 3. Line N is the perpendicular bisector of segment BR. Let U be the midpoint of segment AF. Let Y be the reflection of point E across point C. Point V is the reflection of point Y across line N. Point P is the reflection of point V across segment UA. Find the length of segment PA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center Y and circumradius 10. Point V is the reflection of point E across point B. Let N be the perpendicular bisector of segment EC. Point P is the reflection of point E across point A. Let M be the line through point V perpendicular to line N. Let K be the line through points D and P. Let J be the reflection of line M through point C. Point I is the intersection of lines K and J. Find 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]
Let ABCD be a square with center J and circumradius 1. Let P be the circumcircle of polygon ABCD. Let S be the midpoint of segment BJ. Point M is the midpoint of segment SA. Let T be the inversion of point M in circle P. 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]
Let BCDEFGHIJK be a regular 10-gon with center A and circumradius 5. Let N be the line through points D and B. Line M is the reflection of line N across point K. Point V is the reflection of point F across line M. Line P passes through points V and E. Point L is the intersection of lines P and N. Find the length of segment GL. 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 11. Point M is the reflection of point B across point A. Let N be the reflection of point D across point A. Let Y be the circumcircle of polygon ABCDEF. Point X is the inversion of point N with respect to circle Y. What is the measure of angle XAM, 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 ABCDEG be a regular hexagon with center F and circumradius 9. Let N be the vector from point F to point B. Point Z is the translation of point C by vector N. Line V is the perpendicular bisector of segment FE. Line Q is the reflection of line V across point Z. Point U is the reflection of point B across line Q. Point S is the midpoint of segment CU. Find the length of segment SG. 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 2. Let Y be the angle bisector of angle ABD. Point Z is the midpoint of segment JI. Line P is the reflection of line Y across point M. Point V is the reflection of point J across line P. 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 ABCDEG be a regular hexagon with center F and circumradius 2. Point S is the reflection of point E across point D. Point W is the reflection of point D across segment CS. Point L is the midpoint of segment AW. 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]
ACDEFGHIJKL is a regular 11-gon with center B and circumradius 3. Let Q be the vector from point G to point C. Line R is the angle bisector of angle ICK. Let Y be point I translated by vector Q. Let S be the perpendicular bisector of segment CD. Line O passes through point A perpendicular to line S. Line W is the reflection of line R across line O. Point U is the reflection of point J across line W. Find the distance between points U 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 O and circumradius 12. Let N be the circumcircle of polygon ABCDEF. Line Q is the angle bisector of angle BCD. Let Y be the midpoint of segment AC. Let K be the line through point F perpendicular to line Q. Let Z be the inversion of point Y in circle N. Point S is the reflection of point Z across line K. What is the distance between points Z and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center K and circumradius 1. Line O is the angle bisector of angle EBA. Let I be the perpendicular bisector of segment DE. Let Q be the reflection of line I across line O. Point J is the reflection of point F across line Q. Find the distance between points J and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 10. Line S passes through points E and L. Point T is the reflection of point K across point E. Point P is the reflection of point T across line S. Point Y is the midpoint of segment IP. What is the length of segment KY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center Q and circumradius 11. Let X be the circumcircle of polygon ABCDEFGH. Point V is the midpoint of segment HC. Let T be the inversion of point V in circle X. Let Z be the midpoint of segment TH. What is the distance between points A and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center L and circumradius 11. Line R is the angle bisector of angle GHB. Point P is the reflection of point G across line R. Let V be the midpoint of segment PD. Let Y be the midpoint of segment PH. What is the length of segment VY? 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 12. Line T passes through point D perpendicular to segment AJ. Let W be the line through point E perpendicular to line T. Point P is the reflection of point F across line W. Find the distance between points F and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 9. Let W be the circumcircle of polygon ABDEFGHIJKLM. Circle N is the reflection of circle W across point H. Point O is the inversion of point I with respect to circle N. Line X passes through points K and D. Point Y is the reflection of point O across line X. Point T is the reflection of point B 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]
ABCDFGHI is a regular octagon with center E and circumradius 4. Let X be the perpendicular bisector of segment DF. Line T passes through point F parallel to line X. Let R be the line through point H perpendicular to line T. Line O is the perpendicular bisector of segment EG. Line S is the reflection of line O across line R. Point K is the reflection of point C across line S. Point N is the midpoint of segment KE. Find the distance between points I and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center Q and circumradius 4. Line I is the angle bisector of angle GEA. Line O is the reflection of line I across point B. Point Y is the reflection of point G across point E. Let Z be the line through point E perpendicular to line O. Point J is the reflection of point F across line Z. Find the length of segment JY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKM is a regular dodecagon with center L and circumradius 8. Let Y be the rotation of point C by angle 5pi/12 around point D. Let O be the perpendicular bisector of segment IF. Line V passes through point G parallel to line O. Point S is the intersection of line V and segment EJ. What is the length of segment YS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center W and circumradius 4. Let R be the circumcircle of polygon ABCDEFGH. Circle Z is the reflection of circle R across point F. Let Q be the reflection of circle Z through point W. Point L is the inversion of point H with respect to circle Q. What is the length of segment GL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABDEFG is a regular hexagon with center C and circumradius 5. Let W be the line through points F and C. Let N be the reflection of point E across point B. Let Z be the midpoint of segment AN. Let S be the line through point Z perpendicular to segment BD. Line V is the reflection of line S across line W. Let M be the perpendicular bisector of segment BD. Line O is the reflection of line M across line V. Lines O and W intersect at point K. Let L be the circumcircle of polygon ABDEFG. Point Y is the midpoint of segment DK. Let I be the inversion of point Y in circle L. What is the length of segment IC? 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 6. Line N is the perpendicular bisector of segment BC. Line R passes through point Z perpendicular to line N. Line T passes through point C parallel to line R. Point W is the reflection of point B across point C. Point X is the midpoint of segment WJ. Point S is the reflection of point A across line T. Find the length of segment XS. 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 4. Point L is the midpoint of segment GH. Point Z is the midpoint of segment BC. Point T is the reflection of point C across segment EZ. Let I be the midpoint of segment LT. Find the length of segment FI. 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 4. Point O is the reflection of point D across segment BK. Let T be the perpendicular bisector of segment JK. Line Z passes through point L parallel to line T. Point U is the intersection of line Z and segment BK. Let Q be the midpoint of segment OU. What is the length of segment UQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center Z and circumradius 8. Let V be the perpendicular bisector of segment FD. Line O passes through point D parallel to line V. Point J is the reflection of point Z across line O. Let R be the midpoint of segment CB. Point I is the reflection of point R across point C. What is the distance between points J 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 ABCDEF be a regular hexagon with center T and circumradius 3. Point H is the midpoint of segment DC. Let Q be the circumcircle of polygon ABCDEF. Circle W is the reflection of circle Q across point B. Let X be the inversion of point H in circle W. Find the distance between points X and H. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center U and circumradius 6. Let P be the midpoint of segment AU. Let H be the reflection of point P across point A. Let I be the midpoint of segment DP. What is the distance between points H 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 ABCDEFGH be a regular octagon with center X and circumradius 12. Line Z passes through points X and B. Line T passes through point C perpendicular to line Z. Let I be the midpoint of segment FD. Point W is the reflection of point E across line T. Find the distance between points W and I. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center V and circumradius 3. Let N be the reflection of point A across point F. Let U be the perpendicular bisector of segment AF. Line Q is the reflection of line U across point C. Let K be the line through point N perpendicular to line Q. Let H be the reflection of line K through point D. Lines H and Q intersect at point L. Point T is the reflection of point E across point L. Find the measure of angle TVD, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEF is a regular hexagon with center K and circumradius 3. Let W be the circumcircle of polygon ABCDEF. Let R be the reflection of circle W through point F. Point O is the inversion of point B with respect to circle R. Line U is the angle bisector of angle CBO. Let L be the reflection of circle W across line U. Point Y is the inversion of point A with respect to circle L. What is the length of segment EY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJK is a regular 11-gon with center Q and circumradius 9. Let R be the angle bisector of angle HCA. Point W is the midpoint of segment DA. Line S passes through point F perpendicular to line R. Point N is the reflection of point W across line S. Point O is the reflection of point A across point N. What is the distance between points O and F? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 1. Point Y is the midpoint of segment HL. Let Z be the line through point Y perpendicular to segment EM. Point U is the reflection of point B across line Z. Let O be the perpendicular bisector of segment MB. Line R passes through point A parallel to segment EM. Line Q is the reflection of line O across line R. Point S is the reflection of point U across line Q. What is the length of segment SG? 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 4. Line S is the perpendicular bisector of segment AG. Let O be the circumcircle of polygon ABCDEF. Let X be the reflection of circle O across line S. Point L is the inversion of point D with respect to circle X. Find the distance between points L and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHJ be a regular 9-gon with center I and circumradius 2. Point S is obtained by rotating point C by angle pi about point H. Let L be the circumcircle of polygon ABCDEFGHJ. Let V be the midpoint of segment HA. Line Y passes through points J and S. Point X is the reflection of point V across line Y. Point R is the inversion of point X with respect to circle L. What is the length of segment ER? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGI be a regular octagon with center H and circumradius 12. Line Z is the angle bisector of angle ECD. Line N is the reflection of line Z across point I. Point Y is obtained by rotating point F by angle pi about point C. Let X be the midpoint of segment IA. Let W be the reflection of line N across line Z. Line K is the reflection of line Z across line W. Let T be the reflection of point Y across point I. Point Q is the reflection of point T across line K. What is the distance between points X and Q? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEGHI is a regular octagon with center F and circumradius 8. Let X be the reflection of point E across point I. Let O be the midpoint of segment IH. Point P is obtained by rotating point F by angle 11pi/12 about point I. Line W passes through points H and G. Let U be the midpoint of segment IX. Line K passes through point P parallel to line W. Point J is the reflection of point U across line K. Find the distance between points O and J. 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 M and circumradius 1. Let V be the midpoint of segment IE. Line L passes through point J parallel to segment IE. Line T is the perpendicular bisector of segment DC. Line N is the reflection of line L across point E. Let W be the reflection of line T through point D. Let Z be the line through point M parallel to line N. Lines W and Z intersect at point Y. Point P is the midpoint of segment DC. Point X is the reflection of point P across point V. 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 ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 2. Segments AC and BI intersect at point Q. Point U is the reflection of point D across segment BI. Find the distance between points Q 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 ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 5. Point S is the midpoint of segment AD. Point N is the midpoint of segment LH. What is the distance between points N and S? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEGHIJKLM be a regular dodecagon with center F and circumradius 9. Let R be the angle bisector of angle FAH. Let X be the reflection of point F across point L. Line V is the perpendicular bisector of segment XM. Let Z be the reflection of line V across line R. Point P is the reflection of point L across line Z. What is 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]
Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 6. Point V is the reflection of point A across point I. Let T be the perpendicular bisector of segment EF. Let Y be the circumcircle of polygon ABCDEFGHIJKL. Let Q be the inversion of point V in circle Y. Line M passes through point J perpendicular to line T. Point O is the reflection of point G across line M. Find the length of segment QO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center L and circumradius 10. Let O be the rotation of point B by angle pi around point E. Let Y be the rotation of point O by angle 3pi/4 around point C. Line J passes through point Y parallel to segment LA. Point K is the reflection of point O across segment LA. Let P be the line through point Y perpendicular to line J. Point Q is the reflection of point K across line P. Find 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]
ABCDEFGIJ is a regular 9-gon with center H and circumradius 11. Let X be the line through points E and C. Let L be the reflection of line X through point I. Point O is the reflection of point G across line L. Point R is the midpoint of segment HA. Find 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]
ABCEF is a regular 5-gon with center D and circumradius 3. Point P is the reflection of point F across segment AB. Let Y be the reflection of point F across point D. Point W is the reflection of point Y across segment AB. What is the measure of angle WPA, 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 I and circumradius 6. Line W passes through points D and B. Point Z is the reflection of point G across line W. Point T is the midpoint of segment FA. Point L is the reflection of point Z across point A. 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]
ABCDEFGHIJ is a regular 10-gon with center V and circumradius 3. Let Z be the line through points H and G. Point P is the midpoint of segment AV. Line M passes through point F perpendicular to line Z. Point S is the reflection of point I across line M. Let U be the line through points S and F. Point K is the reflection of point G across line U. 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]
ABCDEF is a regular hexagon with center O and circumradius 6. Let T be the angle bisector of angle CBO. Let K be the circumcircle of polygon ABCDEF. Line I passes through point E perpendicular to line T. Let U be the reflection of circle K through point A. Let P be the inversion of point O in circle U. Let J be the reflection of point P across point B. Point Z is the reflection of point J across line I. Let H be the inversion of point Z in circle K. Find the distance between points H and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center R and circumradius 1. Point Q is the reflection of point D across segment CB. Point I is the reflection of point G across segment CB. Point T is the reflection of point I across point H. 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 ABCDE be a regular 5-gon with center V and circumradius 8. Point H is the midpoint of segment BV. Let I be the angle bisector of angle CAV. Point Q is the midpoint of segment HV. Line N is the reflection of line I across point V. Line J passes through points Q and E. Point O is the intersection of lines N and J. Find the distance between points O and H. 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 2. Line U is the angle bisector of angle FGE. Line S is the perpendicular bisector of segment ID. Let X be the reflection of line U across line S. Point Q is the reflection of point A across line X. What is the measure of angle QFB, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center Y and circumradius 4. Let W be the midpoint of segment CY. Point J is the midpoint of segment HW. Point N is the midpoint of segment JA. Let P be the midpoint of segment JN. Let I be the reflection of point F across point P. What is the distance between points I and C? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 9. Let M be the angle bisector of angle KSC. Let V be the line through point S perpendicular to line M. Let Q be the reflection of point J across point F. Let Z be the reflection of line V through point Q. Point N is the reflection of point G across line Z. Point R is the midpoint of segment HN. Find the length of segment RL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center K and circumradius 2. Point Z is the reflection of point B across point D. Let Q be the midpoint of segment BC. Point U is the reflection of point H across point C. Point T is the reflection of point U across segment CZ. Find the length of segment QT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 5. Let Y be the angle bisector of angle DMB. Line S passes through point L perpendicular to line Y. Point T is the reflection of point J across point I. Point O is the reflection of point D across line S. Point V is the reflection of point O across point H. Find the length of segment TV. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJ is a regular 10-gon with center Y and circumradius 6. Let Q be the midpoint of segment BA. Let N be the reflection of point G across point I. Line L is the angle bisector of angle GYA. Let U be the reflection of line L through point Q. Let S be the reflection of line U through point I. Line K passes through point N perpendicular to line S. Point R is the reflection of point Y across line K. Find the length of segment QR. 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 5. Line W passes through point A perpendicular to segment GJ. Line R is the reflection of line W across point K. Point S is the reflection of point A across line R. What is the length of segment SH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCEF be a regular 5-gon with center D and circumradius 7. Let G be the reflection of point E across point F. Let L be the circumcircle of polygon ABCEF. Circle O is the reflection of circle L across point A. Point T is the midpoint of segment EC. Let K be the inversion of point G in circle O. What is the distance between points T and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGH is a regular octagon with center V and circumradius 8. Line I is the angle bisector of angle GAB. Point R is the reflection of point E across point F. Let S be the reflection of point R across point G. Point J is the reflection of point S across line I. What is the distance between points J and B? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJK is a regular 11-gon with center L and circumradius 2. Line W is the angle bisector of angle KDE. Line M passes through points J and G. Let Q be the reflection of line W across line M. Lines Q and M intersect at point P. Find the distance between points B and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGHIJKL be a regular dodecagon with center P and circumradius 5. Point Y is obtained by rotating point F by angle pi/2 about point L. Point R is the reflection of point Y across point A. Line V is the perpendicular bisector of segment FL. Let N be the line through point P perpendicular to line V. Point S is the reflection of point R across line N. Find the length of segment SJ. 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 X and circumradius 11. Point Q is the midpoint of segment CF. Let H be the midpoint of segment XG. Point T is the midpoint of segment DQ. Let Z be the angle bisector of angle FAT. Line J passes through point H perpendicular to line Z. Point S is the reflection of point B across line J. What is the length of segment AS? 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 V be the midpoint of segment BC. Point N is the reflection of point V across segment GE. Point J is the midpoint of segment CN. Find the distance between points R and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ABCDEFGHIJK is a regular 11-gon with center M and circumradius 5. Line S is the perpendicular bisector of segment MG. Line Z passes through point D perpendicular to segment CE. Line Y passes through point F perpendicular to segment CE. Line X passes through point I parallel to line Z. Line L passes through point M parallel to line S. Line U is the reflection of line X across line Y. Lines U and L intersect at point Q. Find the distance between points C and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEFGH be a regular octagon with center W and circumradius 7. Let Z be the perpendicular bisector of segment FB. Let V be the line through point D parallel to segment EA. Let K be the reflection of line Z through point F. Lines V and K intersect at point P. Let L be the midpoint of segment AP. Point J is the reflection of point L across line Z. What is the distance between points H and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDE be a regular 5-gon with center I and circumradius 1. Point V is the reflection of point E across point I. Let L be the line through points V and A. Point N is the reflection of point I across line L. Let F be the midpoint of segment NI. What is the length of segment EF? 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 2. Let S be the rotation of point E by angle 7pi/12 around point B. Let G be the perpendicular bisector of segment AC. Point K is the midpoint of segment AM. Line L is the reflection of line G across point C. Point T is the reflection of point B across point S. Point Z is the reflection of point C across line L. Point Q is the midpoint of segment TZ. What is the length of segment KQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ABCDEF be a regular hexagon with center S and circumradius 3. Let I be the reflection of point E across point D. Line Z is the perpendicular bisector of segment BD. Point H is the reflection of point I across line Z. Let G be the reflection of point H across point A. What is the length of segment GA? Express your answer as a decimal number rounded to 2 decimal places.

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