[User]
WE = 2. Let X be the rotation of point E by angle 3pi/4 around point W. Point S is obtained by rotating point X by angle 3pi/4 about point E. Let J be the reflection of point W across point E. Let Y be the line through points X and E. Let U be the line through point X perpendicular to line Y. Let V be the midpoint of segment SE. Point M is the midpoint of segment XV. Let F be the midpoint of segment ME. Point I is the reflection of point W across line U. Find the measure of angle IFJ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AW = 7. Point M is obtained by rotating point A by angle 2pi/3 about point W. Point V is obtained by rotating point W by angle pi/12 about point M. Point E is the reflection of point W across point V. Let F be the circumcenter of triangle EMV. Let T be the incenter of triangle EMV. What is the measure of angle TEF, 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 XD = 2. Point Z is the reflection of point X across point D. Point U is obtained by rotating point Z by angle 3pi/4 about point D. Let Y be the perpendicular bisector of segment DZ. Point H is the centroid of triangle XUZ. Point I is the reflection of point H across line Y. Let F be the incenter of triangle XUZ. What is the inradius of triangle IFH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OU = 3. Let G be the rotation of point U by angle pi/4 around point O. Let Y be the incircle of triangle UGO. Point K is the incenter of triangle UGO. Let P be the inversion of point U in circle Y. What is the length of segment KP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let YJ = 4. Let N be the reflection of point Y across point J. Line L is the perpendicular bisector of segment JY. Point O is obtained by rotating point N by angle pi/4 about point Y. Let D be the angle bisector of angle OJN. Let R be the line through point J parallel to line L. Point W is the reflection of point O across line R. Point K is the reflection of point Y across line D. Find the circumradius of triangle KJW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QZ = 12. Let U be the vector from point Z to point Q. Let P be point Q translated by vector U. Let J be the rotation of point Z by angle 11pi/12 around point Q. Let R be the reflection of point Z across point J. Point N is the reflection of point Q across segment RZ. Point I is the midpoint of segment QJ. Find the distance between points P and I. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SX = 1. Point J is obtained by rotating point X by angle pi/12 about point S. Let H be the midpoint of segment SX. Point B is the reflection of point H across point J. Let F be the line through point B perpendicular to segment XJ. Point I is the reflection of point S across line F. 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 FK = 12. Let E be the rotation of point K by angle pi/3 around point F. Let V be the midpoint of segment FE. Point J is the centroid of triangle FEK. Point N is the midpoint of segment JV. Point Y is the reflection of point K across point N. Point P is the reflection of point Y across segment KF. What is the length of segment PY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NX = 8. Point E is obtained by rotating point N by angle 7pi/12 about point X. Let I be the angle bisector of angle NXE. Line F is the reflection of line I across point N. Point H is the reflection of point N across line F. Point J is the midpoint of segment HN. What is the inradius of triangle JHX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XQ = 8. Let G be the rotation of point Q by angle 5pi/6 around point X. Point D is the reflection of point G across point Q. Let J be the circumcenter of triangle DQX. Point N is the incenter of triangle DQX. Point T is the midpoint of segment DQ. What is the inradius of triangle TNJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GV = 2. Point A is obtained by rotating point V by angle pi/3 about point G. Let C be the reflection of point A across point G. Let D be the reflection of point G across point A. Point O is the reflection of point A across point V. What is the inradius of triangle COD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FC = 10. Let L be the rotation of point F by angle pi/3 around point C. Point T is the midpoint of segment LC. Point A is the circumcenter of triangle CFT. Let E be the midpoint of segment TA. Point R is the midpoint of segment TC. Find the measure of angle ERF, 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 SC = 3. Point T is the midpoint of segment CS. Line E is the angle bisector of angle CTS. Line Y is the perpendicular bisector of segment CT. Point H lies on line Y at distance 10 from point S. Let F be the reflection of point T across point H. Let A be the reflection of line Y across line E. Point L is the reflection of point F across line A. Point N is the reflection of point F across segment ST. Point X is the reflection of point N across line E. What is the circumradius of triangle LXN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XK = 2. Point Z is obtained by rotating point K by angle pi/12 about point X. Let S be the perpendicular bisector of segment ZK. Point Y is the reflection of point K across point X. Let N be the line through point K parallel to line S. Let Q be the midpoint of segment YX. Point L is the reflection of point Q across line N. Find the inradius of triangle KLY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RZ = 5. Let N be the midpoint of segment ZR. Let K be the midpoint of segment RN. Point Q is the midpoint of segment ZK. Point I is obtained by rotating point N by angle pi/6 about point Z. Point Y is obtained by rotating point Z by angle pi/4 about point R. Let J be the midpoint of segment QI. Line X is the angle bisector of angle JRQ. Let H be the reflection of line X through point N. Point M is the midpoint of segment KJ. Point L is the reflection of point M across line H. Find the circumradius of triangle YNL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let YC = 5. Point J is the reflection of point C across point Y. Let F be the rotation of point Y by angle 7pi/12 around point C. Line G is the perpendicular bisector of segment JY. Let E be the centroid of triangle YCF. Point U is the orthocenter of triangle YCF. Point A is the reflection of point E across line G. Point B is the reflection of point A across segment CF. What is the measure of angle BUF, 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 CZ = 1. Let D be the rotation of point Z by angle pi/12 around point C. Let W be the angle bisector of angle DZC. Point P is the reflection of point C across line W. Point Y is the reflection of point D across line W. Point J is on line W. Point G is the reflection of point J across segment ZD. Point S is the midpoint of segment DY. Point K is the reflection of point S across segment CZ. Find 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]
ST = 1. Line D passes through points T and S. Point H is obtained by rotating point T by angle 5pi/12 about point S. Let A be the midpoint of segment SH. Point I is the reflection of point H across line D. Point O is the midpoint of segment TA. Let K be the reflection of line D through point I. Point F is the reflection of point T across line K. Let Z be the midpoint of segment SO. What is the measure of angle FTZ, 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 EV = 9. Point Y is obtained by rotating point E by angle 7pi/12 about point V. Circle F is the incircle of triangle EYV. Point R is the centroid of triangle EYV. Let S be the inversion of point Y in circle F. What is the distance between points S 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 FO = 11. Point P is obtained by rotating point O by angle 11pi/12 about point F. Let B be the rotation of point F by angle pi/3 around point O. Line E is the perpendicular bisector of segment PF. Point T is the reflection of point B across line E. Point W is the midpoint of segment PO. Find the inradius of triangle FTW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CO = 3. Point B is obtained by rotating point O by angle pi/2 about point C. Let L be the incenter of triangle COB. Point M is the reflection of point C across segment OB. Point I is the reflection of point L across segment CO. Find the distance between points I and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZN = 8. Point R is obtained by rotating point N by angle pi/3 about point Z. Point K is the orthocenter of triangle RNZ. Let B be the midpoint of segment ZR. Let X be the midpoint of segment RB. What is the measure of angle XKN, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CD = 11. Point S is obtained by rotating point C by angle 3pi/4 about point D. Point R is the incenter of triangle DSC. Let Z be the midpoint of segment DS. Point L is the midpoint of segment ZR. Let K be the midpoint of segment ZS. Find the distance between points K and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BM = 10. Let P be the rotation of point B by angle 11pi/12 around point M. Line Q passes through point P perpendicular to segment BM. Point U is obtained by rotating point P by angle 3pi/4 about point B. Point E is the reflection of point U across segment BM. Point G is the reflection of point M across line Q. Point X is the reflection of point E across line Q. Point Z is the reflection of point M across point G. Let R be the reflection of point X across point Z. Point C is the reflection of point X across segment BM. What is the measure of angle CRG, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GJ = 10. Let N be the rotation of point J by angle 3pi/4 around point G. Let E be the reflection of point G across point N. Point U is the orthocenter of triangle JEN. Let R be the circumcenter of triangle JEN. Let B be the line through point R parallel to segment EN. Point Z is the intersection of segment JE and line B. What is the measure of angle ZNU, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ER = 5. Point P is obtained by rotating point E by angle 7pi/12 about point R. Point C is the reflection of point P across point E. Point V is the incenter of triangle ECR. Point O is the midpoint of segment EC. Find the length of segment VO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RD = 8. Point M is obtained by rotating point D by angle pi/4 about point R. Let H be the rotation of point R by angle 3pi/4 around point D. Let O be the midpoint of segment RH. Point C is the circumcenter of triangle MRO. Point Z is the incenter of triangle MRO. Point W is the incenter of triangle CDZ. Find the length of segment WM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OK = 5. Point S is obtained by rotating point K by angle 3pi/4 about point O. Line M passes through point O perpendicular to segment SK. Let E be the reflection of point K across point O. Let Z be the circumcircle of triangle KOS. Let U be the rotation of point O by angle pi/3 around point E. Let X be the inversion of point U in circle Z. Point J is the reflection of point X across line M. Point P is the orthocenter of triangle KJX. What is the measure of angle XSP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EB = 4. Point J is the reflection of point B across point E. Let Y be the rotation of point J by angle 7pi/12 around point B. Point P is the midpoint of segment EY. Point Z is the circumcenter of triangle JPE. Let Q be the angle bisector of angle JZB. Point M is the midpoint of segment PE. Point D is the reflection of point P across line Q. Find the length of segment MD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let DW = 3. Point R is obtained by rotating point W by angle pi/2 about point D. Let C be the perpendicular bisector of segment RD. Let A be the line through point W perpendicular to line C. Point U is obtained by rotating point D by angle pi/6 about point W. Let G be the circumcenter of triangle RWD. Let H be the midpoint of segment RU. Point E is the reflection of point U across line A. Point S is the reflection of point G across point E. What is the distance between points S and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QJ = 2. Point P is obtained by rotating point Q by angle 3pi/4 about point J. Let C be the circumcenter of triangle QJP. Point I is the midpoint of segment PQ. Let W be the midpoint of segment CP. Find the length of segment IW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SD = 10. Point P is obtained by rotating point S by angle 11pi/12 about point D. Let X be the midpoint of segment PS. Point O is the midpoint of segment PD. Let J be the reflection of point X across point O. Let L be the midpoint of segment PO. Find the measure of angle LJX, 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 RB = 2. Let K be the rotation of point B by angle pi/4 around point R. Let E be the reflection of point R across point B. Point W is the orthocenter of triangle KRE. Let L be the centroid of triangle KRE. What is the distance between points W 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 OC = 4. Point Q is obtained by rotating point O by angle 11pi/12 about point C. Point E is the midpoint of segment QO. Let K be the incircle of triangle CEO. Point N is the inversion of point E with respect to circle K. Find the distance between points Q and N. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WY = 11. Line D is the perpendicular bisector of segment WY. Line H passes through point W perpendicular to line D. Point K is obtained by rotating point W by angle pi/12 about point Y. Point I is obtained by rotating point W by angle pi/12 about point K. Let B be the midpoint of segment IW. Let N be the line through point B parallel to line H. Point T is the reflection of point W across line N. Find the circumradius of triangle TBW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EP = 11. Point T is obtained by rotating point E by angle 11pi/12 about point P. Point J is the centroid of triangle PTE. Line O passes through point E parallel to segment PT. Point A is the reflection of point J across line O. Find the distance between points E 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 UW = 11. Let H be the rotation of point W by angle 2pi/3 around point U. Point M is the incenter of triangle WUH. Let T be the orthocenter of triangle WUH. Point B is the reflection of point M across segment UH. Point O is the reflection of point T across point B. Find the distance between points O and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AK = 7. Point I is obtained by rotating point A by angle pi/2 about point K. Circle Y is the circumcircle of triangle AIK. Let R be the circumcenter of triangle AIK. Circle N is the reflection of circle Y across point R. Let S be the reflection of point K across point I. Let F be the inversion of point A in circle N. Find the length of segment SF. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YS = 4. Point N is the midpoint of segment YS. Let T be the line through point S perpendicular to segment SY. Point V is obtained by rotating point N by angle 5pi/6 about point S. Line A passes through point N parallel to line T. Point O is the reflection of point V across line A. Point I is the midpoint of segment ON. What is the circumradius of triangle NIV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XJ = 9. Point B is obtained by rotating point J by angle 7pi/12 about point X. Let U be the reflection of point B across point J. Point T is the orthocenter of triangle BUX. Point O is the circumcenter of triangle BUX. Point A is the reflection of point U across segment TO. Let D be the midpoint of segment AX. Find the inradius of triangle ABD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let DH = 9. Point Y is obtained by rotating point H by angle pi/6 about point D. Point V is obtained by rotating point H by angle pi/2 about point Y. Let R be the rotation of point Y by angle 5pi/6 around point H. Let T be the line through points H and Y. Line J is the perpendicular bisector of segment DV. Let Q be the midpoint of segment RY. Let O be the reflection of line T through point R. Line U is the reflection of line J across line O. Let I be the midpoint of segment DV. Point M is the reflection of point Q across line U. What is the measure of angle IYM, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PQ = 9. Point K is the midpoint of segment PQ. Let A be the perpendicular bisector of segment KP. Point J lies on line A at distance 8 from point Q. Point R is the midpoint of segment QJ. Point E is the reflection of point K across point J. Let S be the reflection of point R across point P. What is the inradius of triangle ESR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OQ = 4. Let H be the vector from point O to point Q. Let M be the perpendicular bisector of segment OQ. Let J be the line through point O perpendicular to line M. Let U be the rotation of point O by angle pi/6 around point Q. Let L be the reflection of point U across point O. Point X is obtained by rotating point Q by angle 7pi/12 about point O. Point W is the midpoint of segment LQ. Let A be the perpendicular bisector of segment UX. Point F is the translation of point W by vector H. Line V is the reflection of line A across line J. Point T is the reflection of point L across point F. Let R be the midpoint of segment QU. Point N is the reflection of point L across line V. What is the inradius of triangle TNR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TA = 5. Point Y is the midpoint of segment TA. Point M is the midpoint of segment AY. Let B be the rotation of point Y by angle pi/2 around point T. Point W is the reflection of point T across point M. Point H is the midpoint of segment WB. Point D is the reflection of point A across point H. Find the circumradius of triangle BWD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let DL = 1. Let H be the rotation of point D by angle pi/2 around point L. Let K be the midpoint of segment DH. Let S be the reflection of point L across point H. Line F passes through points D and K. Let X be the reflection of line F through point L. Point Y is the reflection of point S across point L. Let V be the reflection of line X across line F. Line Z is the perpendicular bisector of segment HD. Point G is the reflection of point L across line V. Point Q is the reflection of point K across point D. Line B is the angle bisector of angle SDY. Point N is the reflection of point K across line B. Line I passes through point D parallel to line Z. Point O lies on segment QN. Point C is the reflection of point G across line I. Find the measure of angle CSO, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VG = 5. Let X be the rotation of point G by angle 11pi/12 around point V. Let Y be the line through point G perpendicular to segment VX. Point N is the reflection of point V across line Y. Point F is the orthocenter of triangle VXG. Find the length of segment NF. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TK = 10. Point E is the reflection of point K across point T. Let C be the rotation of point T by angle 2pi/3 around point E. Point F is the orthocenter of triangle KEC. Let Q be the line through point E parallel to segment CK. Let N be the reflection of point K across point E. Point Z is the reflection of point N across line Q. What is the circumradius of triangle EFZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QA = 5. Point K is obtained by rotating point Q by angle 5pi/12 about point A. Let Y be the reflection of point A across point Q. Let B be the line through point Q perpendicular to segment QK. Point T is the reflection of point A across line B. Point W is the midpoint of segment QY. Point C is the reflection of point W across segment QK. What is the distance between points C and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OF = 10. Let L be the rotation of point O by angle pi/3 around point F. Let V be the circumcircle of triangle OLF. Let Y be the centroid of triangle OLF. Let E be the angle bisector of angle YFO. Let X be the reflection of line E through point L. Circle G is the reflection of circle V across line X. Let M be the inversion of point O in circle G. Find the inradius of triangle MFY. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BE = 8. Point Z is obtained by rotating point E by angle 3pi/4 about point B. Let Q be the incircle of triangle ZBE. Point R is the orthocenter of triangle ZBE. Point J is the inversion of point R with respect to circle Q. What is the length of segment ZJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IW = 8. Let L be the rotation of point I by angle 3pi/4 around point W. Let F be the rotation of point L by angle pi/2 around point W. Line R passes through points W and F. Let C be the line through point L parallel to line R. Let O be the reflection of point L across point F. Point M is the reflection of point W across line C. Find the measure of angle MIO, 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 PU = 1. Let Z be the rotation of point P by angle 3pi/4 around point U. Let G be the midpoint of segment PU. Let S be the reflection of point Z across point U. Point O is the midpoint of segment PG. Find the circumradius of triangle SZO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let JZ = 7. Let G be the reflection of point Z across point J. Let U be the line through point G perpendicular to segment JZ. Line I is the perpendicular bisector of segment GZ. Let X be the midpoint of segment JZ. Let W be the reflection of line I across line U. Let M be the reflection of line W across line I. Line K is the reflection of line M across line W. Point R is on line U with |RJ| = 8. Let H be the line through points X and R. Point E is the intersection of lines K and H. Point S is the reflection of point G across line H. Point T is the reflection of point E across line W. Find the measure of angle STX, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UR = 4. Let X be the rotation of point R by angle pi/4 around point U. Let F be the perpendicular bisector of segment UX. Line O passes through point R parallel to line F. Let I be the centroid of triangle RUX. Point K is the reflection of point X across line O. What is the distance between points I and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GF = 10. Let S be the rotation of point F by angle pi/6 around point G. Let D be the orthocenter of triangle GSF. Point N is the reflection of point D across segment FG. Let U be the midpoint of segment NF. What is the length of segment UD? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IB = 4. Let E be the rotation of point I by angle pi around point B. Point P is obtained by rotating point I by angle 5pi/6 about point E. Let L be the midpoint of segment PE. Let V be the midpoint of segment LI. Point Y is the reflection of point L across point B. Let S be the line through points Y and V. Point R is the reflection of point E across line S. Find the inradius of triangle RVL. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HA = 10. Point K is obtained by rotating point H by angle 11pi/12 about point A. Let P be the perpendicular bisector of segment AK. Let L be the angle bisector of angle AKH. Point T is the reflection of point H across line P. Point M is the reflection of point A across line L. Let B be the incircle of triangle HMT. Point F is the inversion of point M with respect to circle B. Let Y be the midpoint of segment HM. What is the length of segment YF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XB = 11. Point Y is the reflection of point B across point X. Let I be the midpoint of segment XY. Let F be the midpoint of segment IY. Point Z is obtained by rotating point F by angle 5pi/6 about point X. Point K is the reflection of point Z across point X. Point G is the reflection of point X across point K. Point R is the reflection of point G across segment ZY. Find the measure of angle GYR, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YK = 9. Let L be the rotation of point K by angle pi/6 around point Y. Point F is the incenter of triangle LKY. Line H passes through point F parallel to segment YL. Point E is the intersection of segment KY and line H. What is the measure of angle FEK, 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 KI = 1. Let E be the rotation of point K by angle 5pi/6 around point I. Let Z be the midpoint of segment EI. Point Q is the reflection of point E across point K. Point B is the centroid of triangle KEI. What is the measure of angle ZBQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ON = 5. Point L is obtained by rotating point O by angle pi/12 about point N. Point Y is obtained by rotating point L by angle 5pi/6 about point N. Point Q is the reflection of point L across point N. Point C is the centroid of triangle LNY. Point X is the reflection of point Q across segment YL. Find the measure of angle COX, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LM = 1. Let P be the rotation of point M by angle 11pi/12 around point L. Circle W is the incircle of triangle LPM. Point B is the inversion of point L with respect to circle W. Let E be the midpoint of segment BP. Let Q be the centroid of triangle LPM. Find the length of segment EQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VS = 2. Let I be the rotation of point S by angle 5pi/6 around point V. Point G is the reflection of point I across point V. Point Z is the reflection of point I across segment SG. Point P is the incenter of triangle GIS. What is the measure of angle PZG, 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 UW = 11. Let A be the rotation of point W by angle pi/12 around point U. Let X be the midpoint of segment WU. Point L is obtained by rotating point U by angle 2pi/3 about point A. Point S is obtained by rotating point W by angle pi about point L. Point E is the reflection of point S across point A. Let C be the midpoint of segment XL. What is the inradius of triangle EAC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let Y be the perpendicular bisector of segment BB. Let U be the reflection of point B across point B. Let I be the reflection of point B across point U. Point O is the reflection of point I across line Y. What is the distance between points B and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DB = 11. Let C be the rotation of point B by angle 5pi/6 around point D. Point S is the centroid of triangle BDC. Let Y be the midpoint of segment BD. Point V is the midpoint of segment BY. Point M is the reflection of point S across segment CB. Segments CB and SM intersect at point Q. What is the length of segment QV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JD = 2. Let P be the rotation of point D by angle pi/2 around point J. Let A be the rotation of point P by angle pi/4 around point D. Point V is the midpoint of segment JP. Let X be the reflection of point D across point P. Let O be the line through point P parallel to segment AX. Point Z is the midpoint of segment DV. Point S is the reflection of point J across line O. What is the inradius of triangle SJZ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AF = 8. Point K is obtained by rotating point A by angle pi/3 about point F. Let X be the perpendicular bisector of segment AF. Line S is the reflection of line X across point F. Point E is the reflection of point K across line S. Point Y is the circumcenter of triangle EFK. Find 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]
HA = 2. Point X is obtained by rotating point A by angle pi/4 about point H. Let J be the reflection of point A across point X. Point P is obtained by rotating point X by angle 3pi/4 about point A. Let U be the orthocenter of triangle JXH. Let D be the line through point J parallel to segment XH. Let N be the line through point P parallel to line D. Let Y be the line through points A and U. Lines N and Y intersect at point Z. Point L is the centroid of triangle JXH. Find the distance between points L and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FP = 4. Let R be the rotation of point F by angle 5pi/12 around point P. Point U is the midpoint of segment PR. Point S is the orthocenter of triangle PRF. Let G be the centroid of triangle PRF. Point M is the reflection of point U across point G. Find the length of segment SM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OY = 6. Line D is the perpendicular bisector of segment YO. Point A is the midpoint of segment YO. Point P lies on line D at distance 4 from point O. Let R be the perpendicular bisector of segment AP. Point M is the reflection of point O across line R. Point F is the reflection of point M across point O. What is the inradius of triangle FMY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XK = 10. Let Q be the midpoint of segment KX. Let R be the midpoint of segment QX. Let T be the rotation of point R by angle 3pi/4 around point Q. Let P be the rotation of point K by angle 3pi/4 around point R. Point O is the midpoint of segment TX. Point U is the reflection of point Q across segment KO. Find the inradius of triangle UPQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XW = 7. Point U is obtained by rotating point W by angle pi about point X. Let S be the rotation of point W by angle 5pi/6 around point U. Let Q be the reflection of point X across point S. Point Y is the reflection of point Q across point W. Point T is the midpoint of segment WS. Let H be the midpoint of segment YX. What is the circumradius of triangle TWH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZS = 6. Let X be the rotation of point Z by angle pi/3 around point S. Let D be the circumcircle of triangle ZXS. Point H is the midpoint of segment ZX. Point A is the inversion of point H with respect to circle D. Point C is the midpoint of segment SZ. What is the measure of angle XAC, 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 OS = 1. Let W be the rotation of point S by angle 2pi/3 around point O. Point U is obtained by rotating point S by angle pi/4 about point O. Let N be the midpoint of segment WS. Let E be the rotation of point S by angle 7pi/12 around point U. Point K is the reflection of point U across point E. Line Y is the perpendicular bisector of segment EK. Point G is the reflection of point U across line Y. Find the inradius of triangle ENG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BE = 9. Point S is obtained by rotating point B by angle pi/12 about point E. Point F is obtained by rotating point E by angle 7pi/12 about point B. Let X be the centroid of triangle BES. Circle Z is the circumcircle of triangle BES. Let L be the inversion of point F in circle Z. Point O is the reflection of point L across segment FX. What is the length of segment SO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XB = 2. Let S be the midpoint of segment XB. Let E be the rotation of point B by angle 5pi/6 around point X. Let U be the line through points B and E. Point H is the reflection of point S across line U. Let C be the angle bisector of angle EHS. Point I is the reflection of point X across line C. Point T is the incenter of triangle HIX. Find the distance between points X 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 QL = 8. Point I is the midpoint of segment LQ. Point S is the midpoint of segment LI. Line D is the perpendicular bisector of segment LS. Point W is the reflection of point I across line D. Point V is obtained by rotating point L by angle pi/3 about point S. Let B be the reflection of point V across point Q. Let R be the line through point S parallel to segment IB. Point G is the reflection of point V across line R. Point H is the circumcenter of triangle GVS. What is the measure of angle GHW, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JT = 9. Point M is obtained by rotating point T by angle 5pi/6 about point J. Let R be the reflection of point T across point M. Let V be the rotation of point M by angle pi/3 around point R. Let E be the line through points J and R. Line P passes through point V parallel to segment MJ. Point U is the intersection of lines P and E. What is the length of segment MU? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HX = 10. Let P be the rotation of point X by angle 2pi/3 around point H. Point R is the midpoint of segment PX. Let Q be the midpoint of segment XH. Point E is the orthocenter of triangle RHQ. Find the distance between points P 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 DG = 10. Point W is obtained by rotating point G by angle pi/2 about point D. Point U is the reflection of point D across segment WG. Point S is obtained by rotating point G by angle pi/4 about point U. Point R is the reflection of point S across segment WG. Let N be the reflection of point G across point R. Let B be the midpoint of segment WG. Point F is the reflection of point N across segment WG. What is the measure of angle BNF, 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 VL = 1. Point C is obtained by rotating point L by angle 11pi/12 about point V. Point R lies on segment VL. Let A be the reflection of point R across point L. Let F be the midpoint of segment VL. Point P is the midpoint of segment CF. Line U passes through point P parallel to segment VL. Point D is the reflection of point L across line U. What is the measure of angle ACD, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Point S is the midpoint of segment D52H. Let V be the circumcircle of triangle H. Circle O is the reflection of circle V across point H. Let G be the reflection of circle O through point R. Let U be the inversion of point H in circle G. Point F is the reflection of point H across segment HN. Point Q is the reflection of point S across point F. What is the measure of angle UQF, 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 AX = 6. Let T be the rotation of point A by angle 5pi/12 around point X. Let U be the reflection of point A across point T. Let B be the orthocenter of triangle TUX. Point M is the midpoint of segment BT. Let Q be the centroid of triangle TUX. Find the measure of angle MXQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KZ = 2. Point O is obtained by rotating point K by angle pi/4 about point Z. Let X be the rotation of point Z by angle 5pi/6 around point K. Let A be the midpoint of segment XO. Let U be the perpendicular bisector of segment ZO. Point Q is on line U. Point B lies on line U at distance 4 from point Q. Point T is the midpoint of segment AK. Point L is the midpoint of segment OK. Point N is the midpoint of segment TA. What is the length of segment LN? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XP = 5. Let W be the rotation of point P by angle pi/12 around point X. Let C be the midpoint of segment PW. Let Y be the rotation of point P by angle 2pi/3 around point C. Point Z is the reflection of point Y across segment WP. Let U be the incircle of triangle WPX. Let K be the inversion of point Z in circle U. Point J is the reflection of point W across segment PX. What is the measure of angle WKJ, 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 GW = 2. Let J be the rotation of point G by angle pi around point W. Point F is obtained by rotating point J by angle pi/4 about point W. Let M be the rotation of point J by angle 2pi/3 around point W. Point V is the reflection of point J across point M. Point Q is the midpoint of segment GW. Point B is the orthocenter of triangle GFV. Find the distance between points B and Q. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MQ = 1. Point L is the midpoint of segment MQ. Point P is obtained by rotating point L by angle pi/2 about point M. Point A is obtained by rotating point M by angle 5pi/12 about point P. Point U is the reflection of point M across point P. Line D is the perpendicular bisector of segment UA. Point G is the reflection of point L across line D. Point H is the incenter of triangle UGQ. Find the distance between points U 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 JS = 3. Let L be the rotation of point S by angle 7pi/12 around point J. Let U be the rotation of point J by angle pi/4 around point S. Point X is the reflection of point U across point J. Point M is the incenter of triangle LSU. Line P is the perpendicular bisector of segment LS. Point R is the reflection of point X across line P. What is the measure of angle URM, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VE = 3. Point T is the reflection of point E across point V. Let N be the rotation of point E by angle pi/12 around point V. Point W is obtained by rotating point N by angle pi/6 about point T. Point U is the centroid of triangle VWT. Point J is the midpoint of segment TU. Let B be the orthocenter of triangle VWT. Line I passes through points J and E. Point Q is the reflection of point B across line I. Point Y is the incenter of triangle VWT. Find the length of segment YQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BH = 4. Line L passes through points B and H. Let F be the rotation of point H by angle 5pi/12 around point B. Point P is the reflection of point F across line L. Let E be the angle bisector of angle FPH. Line U is the reflection of line E across line L. Point R is the reflection of point H across line U. Let C be the centroid of triangle RHP. What is the measure of angle CBR, 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 OH = 10. Line L passes through points O and H. Point G is obtained by rotating point O by angle pi/12 about point H. Let C be the line through points O and G. Let K be the reflection of point H across point O. Let R be the midpoint of segment GH. Let A be the line through point G perpendicular to line C. Let B be the reflection of point O across point K. Point N is the reflection of point R across line L. Point E is the reflection of point O across line A. Find the measure of angle EBN, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZF = 8. Let N be the midpoint of segment FZ. Point K is obtained by rotating point F by angle 7pi/12 about point N. Point U is obtained by rotating point F by angle 2pi/3 about point K. Point C is the reflection of point U across point N. Point Q is the reflection of point Z across segment UC. Point V is the orthocenter of triangle ZQC. Let Y be the reflection of point U across point V. Find the measure of angle YKZ, 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 WG = 7. Let D be the rotation of point G by angle 3pi/4 around point W. Let X be the midpoint of segment WD. Point M is the reflection of point W across segment DG. Let O be the circumcenter of triangle DGX. Find the length of segment OM. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XJ = 1. Let Z be the rotation of point J by angle pi/4 around point X. Point R is the centroid of triangle JXZ. Line G passes through point J parallel to segment XZ. Let V be the midpoint of segment XZ. Point S is the reflection of point V across line G. Find the measure of angle RVS, 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 VJ = 11. Point X is obtained by rotating point V by angle 3pi/4 about point J. Let S be the reflection of point J across point X. Let W be the incircle of triangle JVS. Point Y is the inversion of point X with respect to circle W. Find the length of segment YV. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZN = 5. Line B passes through points Z and N. Point J is obtained by rotating point Z by angle 5pi/12 about point N. Let H be the midpoint of segment NZ. Line K passes through point J parallel to line B. Point R is the reflection of point H across line K. Let W be the midpoint of segment ZR. Let F be the circumcenter of triangle HWR. Find the inradius of triangle FNR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VD = 11. Let R be the reflection of point V across point D. Let X be the perpendicular bisector of segment RV. Point W lies on line X at distance 9 from point V. Line K passes through points W and R. Line G is the reflection of line K across point D. Let A be the reflection of line K across line G. Point J is the reflection of point D across line A. What is the measure of angle JVR, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MJ = 5. Point U is obtained by rotating point M by angle pi/2 about point J. Let Y be the line through point M perpendicular to segment MJ. Let C be the rotation of point J by angle pi/4 around point M. Let Q be the angle bisector of angle JCM. Line T is the reflection of line Y across line Q. Point A is the centroid of triangle JUM. Point R is the reflection of point M across line T. Let X be the midpoint of segment AU. 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]
Let BR = 6. Point Q is the midpoint of segment BR. Point V is obtained by rotating point Q by angle 2pi/3 about point R. Let C be the rotation of point R by angle pi/6 around point V. Point D is the midpoint of segment QR. Let T be the rotation of point B by angle 5pi/12 around point D. Line I is the perpendicular bisector of segment BD. Let O be the line through point T perpendicular to segment QR. Let K be the reflection of line O across line I. Point Y is the reflection of point C across line K. What is the distance between points Y and D? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XO = 9. Point M is the midpoint of segment OX. Line B is the perpendicular bisector of segment MO. Point R is the reflection of point X across line B. Point Z is on line B with |ZX| = 5. Point L is the reflection of point X across point Z. Let P be the midpoint of segment RL. Let Y be the reflection of point M across point R. What is the circumradius of triangle YLP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let NB = 5. Let V be the rotation of point N by angle pi/4 around point B. Point A is obtained by rotating point N by angle 5pi/6 about point V. Let K be the line through point N parallel to segment AB. Point X is the reflection of point B across line K. Point O is the orthocenter of triangle VAB. Find the length of segment XO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let DL = 5. Let K be the reflection of point L across point D. Let Y be the reflection of point D across point L. Let Q be the rotation of point D by angle pi/6 around point Y. Line S is the perpendicular bisector of segment KQ. Point A is the reflection of point L across line S. Let W be the circumcenter of triangle YAQ. What is the distance between points W and A? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NA = 11. Point G is obtained by rotating point N by angle pi/3 about point A. Point T is obtained by rotating point G by angle 11pi/12 about point N. Let I be the reflection of point G across point N. Point B is the reflection of point A across segment TI. What is the circumradius of triangle IBA? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AJ = 1. Point Z is the midpoint of segment JA. Point Q is obtained by rotating point J by angle 3pi/4 about point Z. Point F is the midpoint of segment ZQ. Let W be the midpoint of segment QA. Let U be the angle bisector of angle JWQ. Point I is the reflection of point Q across line U. Find the measure of angle FAI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GN = 8. Let P be the midpoint of segment NG. Let Z be the rotation of point P by angle pi/2 around point G. Let W be the reflection of point P across point Z. Point D is the reflection of point P across point N. Point I is the reflection of point D across segment PW. Find the distance between points I 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 BY = 8. Point P is obtained by rotating point Y by angle pi/6 about point B. Point E is obtained by rotating point Y by angle 3pi/4 about point P. Line W is the angle bisector of angle BPY. Let R be the midpoint of segment BY. Point A is the midpoint of segment RY. Point D is the reflection of point E across line W. What is the measure of angle ADP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TF = 11. Point W is obtained by rotating point F by angle 5pi/6 about point T. Point B is the midpoint of segment FW. Point G is the reflection of point F across point T. Let M be the centroid of triangle FBT. Find the distance between points M and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PY = 6. Let D be the rotation of point Y by angle 5pi/6 around point P. Point G is the reflection of point D across point P. Point E is the reflection of point Y across point P. Let U be the midpoint of segment YG. Let W be the reflection of point G across point D. Find the circumradius of triangle EUW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let AV = 12. Let B be the midpoint of segment VA. Let Y be the rotation of point B by angle 5pi/6 around point A. Circle X is the incircle of triangle BYV. Let O be the midpoint of segment YV. Point G is the inversion of point O with respect to circle X. What is the distance between points O and G? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RJ = 4. Point E is obtained by rotating point R by angle 7pi/12 about point J. Let Q be the incenter of triangle EJR. Let S be the midpoint of segment RQ. Let B be the midpoint of segment JR. What is the distance between points S and B? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ME = 12. Point Q is obtained by rotating point E by angle 5pi/12 about point M. Point X is the midpoint of segment QM. Let T be the reflection of point E across point X. Let I be the midpoint of segment QT. Point P is the midpoint of segment QI. What is the circumradius of triangle PMI? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WR = 4. Point P is the midpoint of segment RW. Let L be the line through points W and P. Point X is obtained by rotating point P by angle 3pi/4 about point R. Line Y is the perpendicular bisector of segment WX. Line E passes through point R parallel to line Y. Let Z be the reflection of line L through point X. Point U is the reflection of point X across line Z. Let C be the rotation of point X by angle pi/2 around point P. Point S is the reflection of point C across line E. Let J be the midpoint of segment UC. Let N be the midpoint of segment PJ. Find the length of segment WN. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SI = 2. Point R is obtained by rotating point I by angle pi/3 about point S. Let D be the orthocenter of triangle SRI. Line F is the angle bisector of angle DIS. Point Q is the reflection of point R across line F. Find the length of segment QS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BM = 2. Let F be the rotation of point B by angle pi/6 around point M. Point T is the midpoint of segment MB. Line R is the perpendicular bisector of segment BT. Point O is the reflection of point F across line R. Let I be the midpoint of segment MF. What is the measure of angle TIO, 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 SV = 7. Point F is obtained by rotating point S by angle 3pi/4 about point V. Point B is the reflection of point V across segment FS. Point Y is obtained by rotating point F by angle pi/2 about point B. Line K passes through point Y parallel to segment FS. Point C is the circumcenter of triangle VBY. Point A is the reflection of point C across line K. What is the length of segment AF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UY = 4. Let O be the rotation of point Y by angle 11pi/12 around point U. Let X be the reflection of point U across point Y. Point S is the reflection of point U across segment OX. Let H be the midpoint of segment XY. Find the distance between points H and S. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let UZ = 4. Let W be the rotation of point U by angle 7pi/12 around point Z. Point R is obtained by rotating point W by angle 3pi/4 about point U. Let F be the midpoint of segment UZ. Line D is the angle bisector of angle URF. Let M be the line through point W parallel to line D. Point K is the reflection of point R across line M. Point Q is the midpoint of segment WK. Let P be the midpoint of segment FQ. What is the distance between points P and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XF = 5. Point U is obtained by rotating point X by angle pi/12 about point F. Point L is the midpoint of segment XF. Point P is the reflection of point L across segment XU. Point S is the midpoint of segment PU. Point K is the reflection of point P across segment UF. What is the measure of angle XSK, 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 OA = 6. Let D be the rotation of point A by angle pi/3 around point O. Point X is obtained by rotating point O by angle pi/3 about point A. Point H is the reflection of point D across point O. Let V be the reflection of point O across point H. Let R be the reflection of point H across point A. Find the inradius of triangle VRX. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FG = 12. Point A is obtained by rotating point G by angle 3pi/4 about point F. Let Q be the orthocenter of triangle AGF. Point P is the circumcenter of triangle AGF. Line X passes through point P parallel to segment GF. Point L is the reflection of point A across line X. Point D is the reflection of point L across segment FA. What is the measure of angle DQG, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DK = 7. Point X is the midpoint of segment KD. Line P is the perpendicular bisector of segment DK. Point U is obtained by rotating point X by angle 5pi/6 about point D. Let M be the reflection of line P through point D. Let F be the reflection of line P through point U. Point J is obtained by rotating point X by angle 11pi/12 about point K. Point R is the reflection of point D across point K. Point N is the reflection of point R across line M. Point G is the reflection of point U across point N. Point C is the reflection of point G across line F. Find the measure of angle KCJ, 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 PZ = 12. Point T is obtained by rotating point P by angle 2pi/3 about point Z. Point U is the midpoint of segment ZP. Let W be the midpoint of segment PT. Line N is the perpendicular bisector of segment WU. Point R is the reflection of point P across line N. Find the distance between points U and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GP = 8. Let I be the midpoint of segment GP. Let D be the rotation of point P by angle 2pi/3 around point I. Line B is the perpendicular bisector of segment PD. Point F is the reflection of point G across line B. Point C is the reflection of point D across segment IP. What is the distance between points F 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 KY = 11. Point Z is obtained by rotating point Y by angle 7pi/12 about point K. Point F is the reflection of point Z across segment YK. Let G be the incenter of triangle YKZ. Let J be the reflection of point G across point F. Find the measure of angle JZF, 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 RZ = 11. Let S be the rotation of point Z by angle 5pi/12 around point R. Circle V is the circumcircle of triangle ZRS. Point P is the midpoint of segment RS. Point B is the inversion of point P with respect to circle V. Find the distance between points S 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 PN = 7. Let E be the rotation of point N by angle pi/4 around point P. Point G is the incenter of triangle PEN. Point M is the reflection of point G across segment NP. Let Z be the orthocenter of triangle PEN. Find the distance between points M and Z. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OJ = 9. Let Q be the rotation of point O by angle pi/3 around point J. Point F is the midpoint of segment JO. Let N be the rotation of point J by angle pi/3 around point F. Let H be the rotation of point F by angle pi/2 around point J. Let E be the angle bisector of angle QJH. Let L be the centroid of triangle OHN. Point R is the reflection of point Q across line E. Find the measure of angle RQL, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LB = 4. Let S be the rotation of point L by angle 3pi/4 around point B. Let E be the centroid of triangle SLB. Point X is the incenter of triangle SLB. Let K be the midpoint of segment EX. What is the length of segment KL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PW = 3. Let Z be the midpoint of segment WP. Let A be the rotation of point Z by angle pi/12 around point W. Let I be the midpoint of segment AP. Point V is the centroid of triangle PWA. Point B is the reflection of point V across segment PW. Find the length of segment BI. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let DN = 4. Let V be the rotation of point N by angle pi/12 around point D. Let M be the rotation of point D by angle 5pi/12 around point V. Point K is the orthocenter of triangle NVM. Let C be the reflection of point N across point K. Point Y is the midpoint of segment VM. Let I be the circumcenter of triangle NVM. What is the measure of angle ICY, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BU = 12. Point F is obtained by rotating point B by angle 5pi/6 about point U. Point J is the circumcenter of triangle FUB. Let Z be the incenter of triangle FUB. What is the circumradius of triangle JZB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TH = 4. Let J be the reflection of point H across point T. Let Q be the rotation of point H by angle 7pi/12 around point J. Let A be the midpoint of segment TQ. Line P is the angle bisector of angle AHT. Point C is the midpoint of segment QH. Point W is the reflection of point J across line P. Point S is the midpoint of segment AW. Find the length of segment CS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RJ = 3. Let M be the perpendicular bisector of segment JR. Let C be the rotation of point J by angle 2pi/3 around point R. Let U be the reflection of line M through point C. Let H be the centroid of triangle JCR. Point A is the reflection of point R across line U. What is the length of segment AH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RO = 5. Let W be the rotation of point R by angle 3pi/4 around point O. Line U is the perpendicular bisector of segment RW. Line Y is the perpendicular bisector of segment WO. Let A be the line through point O perpendicular to line U. Line X is the reflection of line Y across line A. Line X and segment WO intersect at point L. Point V is the midpoint of segment OL. Find the distance between points V and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TQ = 8. Let G be the midpoint of segment QT. Point C is obtained by rotating point G by angle 11pi/12 about point T. Line Z is the angle bisector of angle GCQ. Point Y is the reflection of point T across line Z. Line B is the angle bisector of angle CGY. Point F is the reflection of point C across line B. Point S is the circumcenter of triangle YFC. Find the distance between points S and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TG = 1. Point J is obtained by rotating point G by angle pi/12 about point T. Let B be the midpoint of segment GT. Line I is the angle bisector of angle JGT. Point U is the reflection of point B across line I. What is the inradius of triangle UGB? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CA = 9. Point K is obtained by rotating point C by angle pi/6 about point A. Point F is obtained by rotating point A by angle pi/2 about point K. Point W is the circumcenter of triangle KAF. Point L is the midpoint of segment FC. Point P is the midpoint of segment LF. What is the length of segment WP? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let LM = 2. Point Z is the midpoint of segment ML. Let R be the rotation of point L by angle pi/3 around point Z. Line N is the perpendicular bisector of segment ZM. Line X passes through point Z parallel to line N. Point C is the reflection of point R across line X. What is the inradius of triangle ZRC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let KM = 5. Point O is obtained by rotating point M by angle 3pi/4 about point K. Circle T is the incircle of triangle MKO. Point P is the inversion of point K with respect to circle T. Point W is the circumcenter of triangle MKO. What is 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]
CN = 4. Let L be the midpoint of segment CN. Point P is obtained by rotating point L by angle 11pi/12 about point N. Let R be the rotation of point P by angle pi/3 around point L. Let U be the midpoint of segment RL. Point O is the midpoint of segment NU. Point V is the midpoint of segment ON. Point I is the midpoint of segment CP. Find the measure of angle IOL, 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 WT = 8. Point V is the reflection of point T across point W. Point S is the reflection of point V across point T. Point Y is obtained by rotating point T by angle pi/4 about point W. Line R passes through points S and T. Line P is the reflection of line R across point Y. Line L is the reflection of line P across line R. Point B is the reflection of point S across line L. Find the measure of angle YVB, 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 H be the reflection of point J across point F. Point G is the centroid of triangle J. Let D be the reflection of point H across point G. Let I be the midpoint of segment JS. Find the distance between points I 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 GV = 1. Point M is obtained by rotating point V by angle pi/2 about point G. Let K be the reflection of point V across point M. Let F be the reflection of point M across point V. Let R be the midpoint of segment VG. Line J is the perpendicular bisector of segment VG. Point L is the reflection of point K across line J. Find the measure of angle RFL, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NW = 6. Point S is the midpoint of segment WN. Point T is obtained by rotating point W by angle pi/12 about point S. Line G is the perpendicular bisector of segment SN. Line A is the reflection of line G across point S. Point L is the reflection of point T across line G. Point Q is the reflection of point L across line A. What is the measure of angle NLQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LU = 1. Point R is obtained by rotating point L by angle pi/12 about point U. Point Q is the midpoint of segment LU. Point F is the midpoint of segment UQ. Let I be the midpoint of segment RF. Point M is the reflection of point L across segment RF. What is the measure of angle UMI, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WE = 3. Point P is obtained by rotating point E by angle 5pi/12 about point W. Point Z is the reflection of point P across point W. Let V be the rotation of point P by angle pi/4 around point Z. Point F is the midpoint of segment EP. Line D passes through points V and F. Let R be the centroid of triangle FWE. Point X is the reflection of point R across line D. What is the measure of angle EXZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JW = 4. Point K is obtained by rotating point J by angle 7pi/12 about point W. Let D be the rotation of point K by angle pi around point W. Point I is the reflection of point J across point K. Let N be the line through points I and J. Point G is the reflection of point W across line N. Let E be the reflection of point J across point D. Find the measure of angle EGK, 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 XO = 2. Point G is the reflection of point O across point X. Point H is obtained by rotating point G by angle pi/3 about point O. Let E be the midpoint of segment HG. Point S is the midpoint of segment XH. Point L is the reflection of point S across segment HO. What is the distance between points E and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BV = 3. Point Y is the reflection of point V across point B. Point P is obtained by rotating point V by angle 2pi/3 about point Y. Point N is the midpoint of segment VB. Let M be the reflection of point P across point B. Point Q is obtained by rotating point B by angle 7pi/12 about point M. Let K be the reflection of point Y across point P. Let G be the reflection of point N across point K. Let D be the circumcircle of triangle VPQ. Point R is the inversion of point G with respect to circle D. Find the distance between points R and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XY = 6. Point R is obtained by rotating point X by angle 5pi/12 about point Y. Let Z be the rotation of point R by angle pi/3 around point Y. Point B is the reflection of point R across segment YX. Point S is the reflection of point X across point R. Let L be the line through points S and Y. Point T is the reflection of point B across line L. Find the inradius of triangle TZB. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TI = 12. Point S is obtained by rotating point T by angle pi/12 about point I. Let K be the angle bisector of angle ITS. Point Z is the reflection of point S across line K. Point A is the centroid of triangle IZS. Find 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]
Let DJ = 7. Point T is the reflection of point J across point D. DB = 10. Let F be the reflection of point D across point T. Let X be the midpoint of segment BD. Let Z be the midpoint of segment BT. Point Q is the reflection of point Z across segment BD. Find the length of segment XQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZK = 12. Point Y is obtained by rotating point Z by angle 3pi/4 about point K. Point L is the midpoint of segment ZK. Point J is the reflection of point Z across point Y. Let D be the line through point J parallel to segment LZ. Let W be the reflection of point Z across point J. Let R be the reflection of line D through point W. Point G is the reflection of point J across line R. Line U is the perpendicular bisector of segment LZ. Point X is the reflection of point W across line U. Find the measure of angle JGX, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VH = 6. Point X is the reflection of point H across point V. Point U is obtained by rotating point H by angle pi/3 about point X. Let W be the rotation of point H by angle pi/2 around point U. Let J be the reflection of point X across point W. Line Z is the angle bisector of angle UHJ. Point C is obtained by rotating point V by angle 7pi/12 about point X. Let E be the line through point X parallel to line Z. Point Q is the reflection of point U across segment JX. Point K is the reflection of point U across point J. Point M is the midpoint of segment QC. Point F is the reflection of point K across line E. Find the circumradius of triangle MFW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WB = 9. Point Z is the reflection of point W across point B. Point M is obtained by rotating point Z by angle pi/12 about point W. Point G is the midpoint of segment MW. Point Y is the midpoint of segment ZM. What is the circumradius of triangle GBY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let LX = 3. Let C be the rotation of point L by angle pi/3 around point X. Let H be the midpoint of segment CL. Point M is the circumcenter of triangle HXC. Circle Y is the incircle of triangle HXC. Let R be the inversion of point H in circle Y. What is the inradius of triangle MXR? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PC = 2. Let U be the rotation of point P by angle pi/3 around point C. Let E be the midpoint of segment PU. Let Z be the angle bisector of angle UEC. Point M is the reflection of point C across line Z. Let T be the incenter of triangle CPU. Let I be the midpoint of segment PT. Let V be the reflection of point I across point M. Find the length of segment VT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CF = 2. Let S be the rotation of point F by angle 3pi/4 around point C. Point N is the reflection of point S across point C. Let R be the centroid of triangle NFC. Point D is the reflection of point R across segment CN. Find the length of segment DN. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let IW = 4. Point C is the midpoint of segment WI. Point V is obtained by rotating point C by angle 7pi/12 about point I. Point X is the reflection of point V across segment IW. Let R be the incenter of triangle IWV. Let B be the midpoint of segment CR. Find the inradius of triangle XWB. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SE = 8. Point K is obtained by rotating point S by angle 2pi/3 about point E. Let B be the angle bisector of angle EKS. Point A is the midpoint of segment KE. Point N is the midpoint of segment AE. Point D is the midpoint of segment SA. Point Y is the reflection of point D across line B. Find the circumradius of triangle NYS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WF = 4. Let S be the rotation of point F by angle 5pi/12 around point W. Point R is obtained by rotating point W by angle 3pi/4 about point S. Let D be the midpoint of segment WF. Point B is the reflection of point W across segment RD. Point T is the orthocenter of triangle RDS. Point C is the midpoint of segment FT. What is the distance between points T and C? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
JC = 9. Let Q be the rotation of point J by angle pi/6 around point C. Point D is the incenter of triangle CJQ. Point Y is the reflection of point D across point J. Point V is the reflection of point Y across segment CJ. Let O be the reflection of point V across point Q. What is the measure of angle OVC, 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 VE = 1. Point X is obtained by rotating point V by angle pi/6 about point E. Point A is the reflection of point X across point E. Point B is the midpoint of segment VX. Let S be the reflection of point V across point A. Line U passes through point B perpendicular to segment VX. Point Q is the reflection of point S across line U. What is the inradius of triangle BSQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let K be the orthocenter of triangle O. Point Q is the midpoint of segment IK. Point D is the reflection of point I across segment PW36. Let J be the midpoint of segment DQ. Find 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]
TY = 9. Point C is obtained by rotating point T by angle 2pi/3 about point Y. Let J be the incenter of triangle CYT. Point U is the orthocenter of triangle CYT. Let Q be the midpoint of segment CU. Let I be the reflection of point Q across point J. Find the distance between points T and I. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let UW = 1. Let T be the midpoint of segment WU. Let I be the perpendicular bisector of segment TW. Let G be the reflection of line I through point W. Point X is on line G with |XT| = 11. Point K is the reflection of point X across line I. Let A be the midpoint of segment UT. Let F be the incenter of triangle KWU. What is the measure of angle AKF, 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 AR = 10. Let N be the rotation of point A by angle pi/12 around point R. Let T be the rotation of point A by angle pi/2 around point R. Let V be the perpendicular bisector of segment NR. Let O be the incircle of triangle RNA. Circle W is the reflection of circle O across line V. Let H be the midpoint of segment NT. Let S be the inversion of point A in circle W. Find the measure of angle TSH, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NH = 1. Point W is obtained by rotating point H by angle 5pi/12 about point N. Let P be the rotation of point N by angle 5pi/12 around point H. Let J be the incenter of triangle PNW. Let B be the incircle of triangle PNW. Point U is the inversion of point H with respect to circle B. Let R be the midpoint of segment JU. Point G is the midpoint of segment PN. Let Z be the midpoint of segment NR. Find the measure of angle ZJG, 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 VM = 12. Let D be the rotation of point V by angle pi/6 around point M. Point S is the reflection of point D across point V. Point N is the orthocenter of triangle MSV. Point B is the midpoint of segment SM. Point Y is the midpoint of segment BV. Find the measure of angle YBN, 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 MQ = 11. Point Z is obtained by rotating point Q by angle pi/6 about point M. Let T be the circumcenter of triangle ZQM. Let G be the angle bisector of angle TQZ. Point S is the reflection of point M across segment ZQ. Point O is the reflection of point S across line G. Find the length of segment OZ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LZ = 3. Point M is obtained by rotating point Z by angle 7pi/12 about point L. Let C be the orthocenter of triangle ZLM. Let Q be the vector from point Z to point C. Point S is the translation of point M by vector Q. Point F is the midpoint of segment SM. Let J be the incenter of triangle ZLM. Circle E is the incircle of triangle ZLM. Let K be the reflection of circle E through point F. Let A be the inversion of point S in circle E. Point G is the inversion of point J with respect to circle K. Find the distance between points G and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IL = 11. Line V passes through point I perpendicular to segment LI. Point Q is the reflection of point L across line V. Point C lies on line V at distance 5 from point L. Point B is the reflection of point C across point Q. Point W is the reflection of point C across point I. Let Y be the midpoint of segment WL. What is the distance between points W and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WN = 11. Let K be the rotation of point N by angle 5pi/12 around point W. Let S be the line through points N and K. Let F be the line through point W perpendicular to line S. Let A be the midpoint of segment NK. Line R is the reflection of line F across point K. Point M is the reflection of point N across point W. Point O is the reflection of point W across line R. Let H be the perpendicular bisector of segment MN. Point X is the reflection of point A across line H. What is the measure of angle WOX, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
HC = 11. Let S be the rotation of point H by angle 3pi/4 around point C. Point J is obtained by rotating point H by angle 5pi/12 about point S. Point M is the reflection of point H across segment CS. Let K be the circumcenter of triangle CJH. Let A be the midpoint of segment CJ. Point L is the midpoint of segment AK. Find the length of segment ML. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TP = 7. Line C is the perpendicular bisector of segment PT. Let L be the reflection of line C through point P. Point D is on line L with |DT| = 7. Point M is the reflection of point D across point P. Point V is the reflection of point P across segment TM. Point Y is the midpoint of segment DT. What is the distance between points V and Y? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LV = 7. Point G is obtained by rotating point L by angle pi/12 about point V. Point X is obtained by rotating point L by angle pi/3 about point V. Let P be the reflection of point X across point L. Let F be the line through point G parallel to segment VX. Line Q passes through point X perpendicular to line F. Point U is the reflection of point V across line Q. Point J is the reflection of point P across segment VX. Let S be the midpoint of segment XG. Find the circumradius of triangle JUS. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TK = 9. Let S be the reflection of point K across point T. Let X be the vector from point T to point K. Point B is obtained by rotating point K by angle 11pi/12 about point T. Point G is obtained by rotating point K by angle 5pi/12 about point S. Point W is the translation of point G by vector X. Let Z be the midpoint of segment BG. Point P is the reflection of point T across point S. Point L is the midpoint of segment PW. Point A is the circumcenter of triangle PWS. Let U be the midpoint of segment ZL. Find the measure of angle UKA, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
DJ = 9. Let A be the rotation of point J by angle pi/12 around point D. Point I is the reflection of point D across point J. Point M lies on segment JD. Point P is the reflection of point M across segment DA. Find the measure of angle IDP, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZB = 7. Point L is obtained by rotating point B by angle 11pi/12 about point Z. Let X be the circumcircle of triangle ZBL. Let Q be the incenter of triangle ZBL. Let Y be the centroid of triangle ZBL. Point J is the reflection of point Y across segment ZB. Let G be the inversion of point J in circle X. What is the length of segment GQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QW = 7. Let O be the perpendicular bisector of segment WQ. Line P is the reflection of line O across point W. Point J is the reflection of point Q across line P. Point T lies on line P at distance 1 from point J. Point A is the circumcenter of triangle QTW. Point G is the incenter of triangle QTW. What is the circumradius of triangle AQG? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZR = 2. Let O be the midpoint of segment ZR. Point Y is the reflection of point O across point R. Let W be the rotation of point Y by angle pi/4 around point Z. Point L is obtained by rotating point O by angle pi/3 about point Z. Point K is the midpoint of segment WY. Let P be the reflection of point O across point L. Let I be the midpoint of segment KP. What is the inradius of triangle OIW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VJ = 4. Let B be the reflection of point J across point V. Let Z be the rotation of point B by angle 5pi/6 around point J. Let Q be the line through points B and Z. Point G is the reflection of point V across point J. Line W passes through point V parallel to line Q. Let X be the midpoint of segment GJ. Point T is the reflection of point V across line Q. Point C is the reflection of point G across line W. What is the circumradius of triangle CXT? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OZ = 8. Point Q is the midpoint of segment OZ. Point W is obtained by rotating point Z by angle 5pi/6 about point O. Let K be the midpoint of segment WQ. Line A is the perpendicular bisector of segment KQ. Line F is the perpendicular bisector of segment ZQ. Let J be the rotation of point Z by angle 3pi/4 around point K. Line H passes through point J parallel to line A. Line C passes through point J perpendicular to line H. Line E is the reflection of line F across line C. Point L is the reflection of point K across line E. What is the measure of angle OWL, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XI = 12. Let T be the midpoint of segment IX. Point D is obtained by rotating point T by angle 11pi/12 about point X. Point H is the midpoint of segment DT. Line J is the angle bisector of angle IHX. Point M is the reflection of point T across line J. Let Y be the centroid of triangle HMI. Find the length of segment YD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UO = 6. Let A be the rotation of point O by angle pi/2 around point U. Let E be the rotation of point A by angle 5pi/12 around point U. Let J be the midpoint of segment OU. Point X is the incenter of triangle UAO. Point K is the reflection of point X across segment EA. Find the measure of angle KJX, 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 PT = 7. Let S be the midpoint of segment TP. Point X is the midpoint of segment PS. Point K is the reflection of point S across point P. Point Q is obtained by rotating point K by angle 3pi/4 about point X. Let Z be the reflection of point Q across point X. Point A is the reflection of point P across point Z. Point G is the orthocenter of triangle QZS. Find the distance between points A and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
TV = 8. Point R is obtained by rotating point V by angle pi/2 about point T. Point Y is the midpoint of segment TR. Point C is the reflection of point V across point Y. Point A is the reflection of point C across segment RV. What is the measure of angle TAC, 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 XO = 10. Point P is obtained by rotating point X by angle pi/2 about point O. Let W be the line through points O and P. Point G is the reflection of point X across line W. Let L be the midpoint of segment OG. Let T be the reflection of point G across point P. What is the measure of angle PLT, 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 ON = 7. Let E be the rotation of point O by angle 3pi/4 around point N. Point K is the midpoint of segment NE. Let T be the rotation of point K by angle pi/3 around point N. Point Q is the reflection of point T across point O. Line H is the perpendicular bisector of segment TQ. Point W is the reflection of point E across line H. Find the distance between points O and W. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YW = 4. YP = 4. Let U be the perpendicular bisector of segment PW. Let S be the reflection of point Y across point W. Point N is the reflection of point S across line U. Let G be the midpoint of segment PY. Find the distance between points N and G. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YU = 3. Point O is obtained by rotating point Y by angle pi/4 about point U. Let H be the midpoint of segment YO. Point J is the orthocenter of triangle OUY. Let G be the midpoint of segment YH. What is the distance between points J and G? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WF = 7. Let J be the rotation of point F by angle 11pi/12 around point W. Point Z is the reflection of point J across segment WF. Point S is the circumcenter of triangle WJZ. Let O be the midpoint of segment ZS. Let U be the incenter of triangle WJZ. Find the distance between points U and O. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CT = 8. Point S is obtained by rotating point T by angle pi/4 about point C. Point Y is the midpoint of segment CS. Point V is the reflection of point T across segment CS. Let G be the reflection of point V across point Y. Let I be the circumcenter of triangle TVG. Find the distance between points I and V. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YH = 9. Point A is the midpoint of segment HY. Point X is obtained by rotating point A by angle 5pi/6 about point H. Point T is the reflection of point X across point A. Line I is the angle bisector of angle ATH. Point N is the reflection of point Y across line I. Point W is the midpoint of segment NH. What is the length of segment XW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YF = 1. Point J is the midpoint of segment FY. Let W be the rotation of point Y by angle 2pi/3 around point F. Point L is the reflection of point F across point W. Let B be the midpoint of segment JF. Let K be the reflection of point Y across point B. Find the circumradius of triangle LKJ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YO = 7. Point B is obtained by rotating point O by angle 5pi/12 about point Y. Point V is the midpoint of segment BY. Let K be the reflection of point B across point O. Find the circumradius of triangle KOV. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZI = 3. Let U be the reflection of point Z across point I. Let M be the rotation of point U by angle 11pi/12 around point I. Let N be the incenter of triangle IMZ. Let J be the circumcenter of triangle IMZ. Point Q is the reflection of point N across segment MZ. Point G is the reflection of point Q across segment IM. What is the distance between points J and G? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BW = 3. Line F passes through points B and W. Point Q is obtained by rotating point B by angle 3pi/4 about point W. Let R be the rotation of point B by angle 5pi/12 around point Q. Point A is the reflection of point R across line F. Point P is the reflection of point Q across point R. Line M is the angle bisector of angle WPQ. Point U is the reflection of point B across line M. Point Z is the reflection of point R across line M. Let K be the reflection of point W across point Z. Find the measure of angle KUA, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let UI = 12. Let G be the rotation of point U by angle 11pi/12 around point I. Let F be the line through points G and I. Let Q be the reflection of point G across point I. Point W is the reflection of point U across line F. What is the inradius of triangle WUQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XK = 2. Point Q is obtained by rotating point X by angle 11pi/12 about point K. Point E is the midpoint of segment QK. Let V be the centroid of triangle XQK. Point G is the reflection of point E across segment KX. What is the length of segment GV? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RO = 2. Point P is obtained by rotating point R by angle pi/2 about point O. Let H be the incenter of triangle RPO. Circle L is the incircle of triangle RPO. Point W is the inversion of point R with respect to circle L. Let J be the midpoint of segment HP. Find the length of segment JW. Express your answer as a decimal number rounded to 2 decimal places.

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