[User]
VQ = 6. Let P be the midpoint of segment QV. Point L is the reflection of point V across point Q. Let C be the perpendicular bisector of segment QV. Line X passes through point V parallel to line C. Line O is the reflection of line X across line C. Point D is the reflection of point P across line X. Line K is the reflection of line O across point V. Let M be the midpoint of segment PV. Let E be the rotation of point V by angle 5pi/6 around point M. Let F be the line through points E and D. Point I is the reflection of point L across line F. Point Y is the reflection of point I across line K. What is the circumradius of triangle MIY? Express your answer as a decimal number rounded to 2 decimal places.

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Let AO = 10. Let S be the rotation of point O by angle 3pi/4 around point A. Point M is the midpoint of segment SA. Let V be the circumcenter of triangle SOA. Point P is the reflection of point M across segment OA. What is the measure of angle SVP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let LX = 12. Let E be the rotation of point L by angle 11pi/12 around point X. Let A be the rotation of point L by angle 3pi/4 around point E. Point V is the midpoint of segment XL. Point S is the circumcenter of triangle EAX. What is the distance between points S and V? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
NY = 12. Let P be the rotation of point N by angle 11pi/12 around point Y. Point H is the orthocenter of triangle NYP. Let W be the incircle of triangle NYP. Let U be the circumcenter of triangle NYP. Point I is the midpoint of segment PN. Let D be the reflection of circle W through point I. Point J is the midpoint of segment YP. Point Q is the inversion of point J with respect to circle D. Let C be the midpoint of segment UH. What is the length of segment CQ? Express your answer as a decimal number rounded to 2 decimal places.

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Let FM = 10. Let R be the reflection of point F across point M. Point C is obtained by rotating point R by angle 3pi/4 about point F. Point P is obtained by rotating point M by angle 11pi/12 about point R. Point O is the reflection of point F across point C. Line V is the perpendicular bisector of segment MC. Let G be the reflection of line V through point C. Point Z is the reflection of point O across line G. What is the circumradius of triangle PZF? Express your answer as a decimal number rounded to 2 decimal places.

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Let XQ = 2. Let J be the rotation of point Q by angle 3pi/4 around point X. Point F is the incenter of triangle QJX. Point Y is the reflection of point F across point Q. Point K is the centroid of triangle QJX. Let W be the midpoint of segment QJ. What is the circumradius of triangle KYW? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let VE = 11. Point G is obtained by rotating point V by angle pi/12 about point E. Let U be the circumcenter of triangle EGV. Point N is the reflection of point U across segment EG. What is the inradius of triangle GNU? Express your answer as a decimal number rounded to 2 decimal places.

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NR = 11. Line Z is the perpendicular bisector of segment NR. Point M is the reflection of point N across point R. Point B is on line Z with |BR| = 5. Point W is the reflection of point M across point N. Line E is the angle bisector of angle MNB. Line Y passes through point M perpendicular to line E. Line I passes through point B perpendicular to line Y. Point T is the reflection of point M across line I. What is the circumradius of triangle WBT? Express your answer as a decimal number rounded to 2 decimal places.

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Let IL = 3. Point N is obtained by rotating point I by angle 7pi/12 about point L. Point A is the midpoint of segment NI. Let Z be the midpoint of segment AL. Let F be the midpoint of segment NA. Find the length of segment ZF. Express your answer as a decimal number rounded to 2 decimal places.

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NR = 8. Let W be the reflection of point N across point R. Let C be the rotation of point R by angle 7pi/12 around point W. Let F be the vector from point N to point R. Let P be the midpoint of segment RN. Point S is the translation of point P by vector F. Let V be the angle bisector of angle NCP. Point D is the midpoint of segment SC. Point K is the reflection of point D across line V. Let O be the midpoint of segment PK. Let H be the midpoint of segment WC. What is the inradius of triangle CHO? Express your answer as a decimal number rounded to 2 decimal places.

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ZQ = 9. Point V is the midpoint of segment QZ. Line S passes through points Q and V. Line W passes through point V perpendicular to line S. Point A lies on line W at distance 7 from point Q. Let D be the midpoint of segment VA. Point P is the reflection of point Q across point A. Point I is the reflection of point D across line S. Let M be the orthocenter of triangle PAZ. What is the distance between points I and M? Express your answer as a decimal number rounded to 2 decimal places.

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OK = 7. Let Q be the rotation of point K by angle 7pi/12 around point O. Let T be the reflection of point Q across point O. Point U is the reflection of point K across point T. Point H is obtained by rotating point K by angle pi/3 about point O. Let W be the incircle of triangle KTH. Circle M is the reflection of circle W across point K. Point X is the inversion of point K with respect to circle M. What is the distance between points U and X? Express your answer as a decimal number rounded to 2 decimal places.

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ED = 8. Point X is obtained by rotating point D by angle pi/6 about point E. Line G passes through points X and D. Point S is the reflection of point E across line G. Let L be the orthocenter of triangle EXS. Let I be the midpoint of segment SL. Find the length of segment SI. Express your answer as a decimal number rounded to 2 decimal places.

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ZE = 5. Point Y is obtained by rotating point Z by angle 2pi/3 about point E. Point P is obtained by rotating point E by angle pi/12 about point Z. Let H be the rotation of point Y by angle pi/3 around point Z. Let F be the incenter of triangle HZP. Let A be the incircle of triangle HZP. Point W is the inversion of point Z with respect to circle A. What is the measure of angle EFW, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let PJ = 7. Let L be the rotation of point J by angle pi/6 around point P. Point A is the midpoint of segment JP. Point V is the incenter of triangle JPL. Point D is the orthocenter of triangle JPL. Line T is the perpendicular bisector of segment VD. Point N is the reflection of point A across line T. Find the distance between points N and A. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
TL = 6. Let D be the reflection of point T across point L. Let H be the rotation of point T by angle 2pi/3 around point D. Let S be the rotation of point T by angle pi/12 around point L. Point B is the reflection of point S across point H. Let X be the midpoint of segment BL. Point I is the reflection of point S across point B. Let V be the reflection of point L across point S. What is the circumradius of triangle XIV? Express your answer as a decimal number rounded to 2 decimal places.

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KE = 7. Let T be the rotation of point E by angle pi/4 around point K. Let D be the midpoint of segment TK. Point P is the midpoint of segment ET. Let A be the perpendicular bisector of segment ED. Let S be the line through point E perpendicular to line A. Point X is the reflection of point P across segment TK. Point M is the reflection of point X across line S. What is the distance between points M and X? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let RC = 7. Let S be the line through points R and C. Let U be the midpoint of segment CR. Point X is obtained by rotating point C by angle 5pi/6 about point R. Let A be the midpoint of segment XR. Point Z is the reflection of point X across line S. Let L be the midpoint of segment ZU. Let P be the line through point A parallel to line S. Let M be the midpoint of segment ZC. Point J is on line P. What is the measure of angle LMJ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ZX = 12. Line C is the perpendicular bisector of segment XZ. Let T be the reflection of point X across point Z. Line E passes through point X parallel to line C. Let B be the rotation of point T by angle pi/3 around point Z. Let J be the midpoint of segment TZ. Let Y be the reflection of line E through point J. Point Q is the reflection of point X across line Y. Find the circumradius of triangle TBQ. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
LK = 1. Point A is obtained by rotating point L by angle 7pi/12 about point K. Let T be the midpoint of segment AL. Let S be the centroid of triangle LAK. Let W be the incircle of triangle LAK. Point D is the inversion of point S with respect to circle W. What is the distance between points T and D? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
KA = 10. Point E is obtained by rotating point A by angle 2pi/3 about point K. Let O be the midpoint of segment EA. Point L is obtained by rotating point A by angle pi/6 about point O. Point D is the reflection of point E across segment AK. Point H is the midpoint of segment OD. Point V is the reflection of point L across segment DE. Point U is the midpoint of segment VH. Point Z is the midpoint of segment AH. Find the distance between points U and Z. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
XE = 6. Point Q is obtained by rotating point X by angle 5pi/6 about point E. Point I is the incenter of triangle EQX. Point F is the midpoint of segment EQ. Let J be the perpendicular bisector of segment XE. Let M be the midpoint of segment QX. Let C be the midpoint of segment IM. Point A is the reflection of point F across line J. Find the length of segment AC. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let AM = 4. Point H is the midpoint of segment AM. Let K be the reflection of point A across point M. Let B be the perpendicular bisector of segment KA. Point C lies on line B at distance 1 from point A. Let R be the reflection of line B through point A. Point X lies on line R. Let Q be the circumcenter of triangle CXA. What is the measure of angle HQX, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let PN = 1. Let O be the reflection of point P across point N. Point X is obtained by rotating point N by angle 7pi/12 about point P. Point I is the midpoint of segment XP. Let Z be the circumcenter of triangle OIX. Point C is the midpoint of segment ZI. What is the length of segment CZ? Express your answer as a decimal number rounded to 2 decimal places.

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GP = 3. Let C be the midpoint of segment PG. Let W be the line through point P perpendicular to segment GC. Point E is obtained by rotating point P by angle pi/12 about point G. Point V is the reflection of point E across line W. Let R be the line through points E and C. Let J be the rotation of point E by angle pi around point P. Line A passes through point P perpendicular to line R. Point M is the reflection of point E across line A. Point D is the midpoint of segment EV. Find the measure of angle JDM, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
VE = 4. Let N be the rotation of point V by angle pi/12 around point E. Point W is the midpoint of segment EN. Let J be the midpoint of segment WE. Let Y be the angle bisector of angle JVN. Let L be the line through point W perpendicular to line Y. Let X be the line through point W perpendicular to line L. Point Z is the reflection of point V across line X. Find the circumradius of triangle JEZ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZH = 3. Let U be the rotation of point Z by angle 7pi/12 around point H. Let R be the incenter of triangle UHZ. Let Q be the midpoint of segment HR. Point A is the midpoint of segment ZQ. Let Y be the line through point R parallel to segment HZ. Point N is the intersection of segment UH and line Y. Find the distance between points A and N. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
AP = 2. Let Q be the rotation of point P by angle 5pi/6 around point A. Let D be the rotation of point Q by angle 11pi/12 around point A. Point R is the reflection of point Q across point D. Let W be the incenter of triangle QAP. Let S be the orthocenter of triangle QAP. Line K passes through points W and S. Point J is the reflection of point A across segment PQ. Point T is the reflection of point R across line K. Find the distance between points T and J. Express your answer as a decimal number rounded to 2 decimal places.

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Let ZV = 8. Line D passes through points V and Z. Point L is the midpoint of segment ZV. Point A lies on line D at distance 12 from point V. Point J is obtained by rotating point Z by angle 2pi/3 about point V. Point X is the midpoint of segment AJ. Let P be the midpoint of segment JX. Point I is the midpoint of segment ZX. Let O be the midpoint of segment JP. Let B be the midpoint of segment JZ. Point T is the reflection of point L across point I. Find the measure of angle BOT, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let MQ = 6. Let K be the rotation of point M by angle pi/4 around point Q. Let W be the vector from point K to point M. Let E be the reflection of point Q across point K. Point L is the orthocenter of triangle QKM. Point I is the circumcenter of triangle QKM. Let C be the midpoint of segment QK. Let U be the line through point E perpendicular to segment MQ. Point S is the translation of point M by vector W. Let P be the line through point M parallel to line U. Point J is the reflection of point L across line P. Line N is the perpendicular bisector of segment EJ. Let T be the reflection of line N across line P. Point Z is the reflection of point C across line T. What is the measure of angle IZS, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let NY = 7. Let E be the midpoint of segment NY. Let L be the midpoint of segment EY. Line S is the perpendicular bisector of segment NE. Point G is on line S with |GE| = 7. Let W be the reflection of point L across point N. Point O is the midpoint of segment EG. Let C be the midpoint of segment GN. Let Q be the midpoint of segment WO. What is the circumradius of triangle QCL? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
JP = 11. Let L be the line through points J and P. Point W is obtained by rotating point J by angle 5pi/12 about point P. Point E is obtained by rotating point J by angle pi/12 about point W. Point R is the reflection of point W across line L. Point Y is the reflection of point P across segment EW. Find the inradius of triangle RYW. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let WT = 5. Point A is obtained by rotating point W by angle 7pi/12 about point T. Point I is the reflection of point A across segment TW. Line V passes through points I and T. Point N is the intersection of segment WA and line V. What is the distance between points N and W? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
VU = 7. Let J be the vector from point V to point U. Let S be point U translated by vector J. Let W be the rotation of point U by angle 7pi/12 around point S. Point T is the reflection of point W across point V. Let O be the midpoint of segment UV. Circle Y is the circumcircle of triangle OTW. Let X be the orthocenter of triangle OTW. Point B is the inversion of point X with respect to circle Y. Find the measure of angle UWB, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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LV = 9. Point T is obtained by rotating point L by angle 3pi/4 about point V. Line W is the perpendicular bisector of segment TL. Point F is the midpoint of segment VL. Point S is the reflection of point F across line W. Point X is the reflection of point F across point T. Let I be the reflection of point S across point F. Find the measure of angle XSI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let AO = 6. Let W be the perpendicular bisector of segment OA. Let L be the rotation of point A by angle pi around point O. Let I be the midpoint of segment AO. Let U be the perpendicular bisector of segment LI. Point B is obtained by rotating point I by angle pi/4 about point A. Line Q is the reflection of line U across line W. Point P is the reflection of point L across line Q. Point M is obtained by rotating point B by angle 11pi/12 about point I. Let V be the midpoint of segment OB. Point F is the incenter of triangle MBP. Let R be the orthocenter of triangle MBP. Find the measure of angle VFR, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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NQ = 1. Let T be the rotation of point N by angle pi/6 around point Q. Line S is the perpendicular bisector of segment QT. Line Z is the reflection of line S across point Q. Point Y is the reflection of point T across line Z. What is the inradius of triangle NYT? Express your answer as a decimal number rounded to 2 decimal places.

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BL = 4. Let Y be the rotation of point L by angle pi/12 around point B. Let I be the orthocenter of triangle BYL. Let X be the midpoint of segment LB. Let K be the reflection of point Y across point B. What is the measure of angle IKX, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let BF = 7. Let V be the reflection of point F across point B. Point D is obtained by rotating point V by angle 7pi/12 about point F. Point L is the reflection of point B across point D. Point I is the midpoint of segment BV. Let G be the reflection of point L across point I. What is the circumradius of triangle GDL? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let RU = 7. Let Y be the rotation of point U by angle pi/3 around point R. Point M is obtained by rotating point Y by angle 5pi/6 about point R. Point K is the reflection of point R across point U. Let H be the circumcircle of triangle MRY. Let N be the inversion of point K in circle H. What is the circumradius of triangle NKY? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
JT = 9. Point R is the reflection of point T across point J. Line Y is the angle bisector of angle RTJ. Let E be the rotation of point J by angle 3pi/4 around point R. Point X is the midpoint of segment ER. Line C is the perpendicular bisector of segment XR. Point F lies on line Y. Point D is the reflection of point J across line C. Point A is the midpoint of segment ED. Find the distance between points A and D. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let QO = 3. Point Y is obtained by rotating point O by angle pi/12 about point Q. Let V be the midpoint of segment QO. Let H be the midpoint of segment YO. Let G be the rotation of point Y by angle pi/4 around point V. Line W is the angle bisector of angle GQY. Point J is the reflection of point H across point Q. Let M be the line through points O and G. Point K is the reflection of point Y across line M. Line P passes through points K and Y. Line B is the reflection of line W across line P. Point Z is the intersection of lines B and P. Point A is the midpoint of segment GJ. What is the measure of angle GZA, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let YT = 4. Let D be the rotation of point Y by angle 7pi/12 around point T. Point X is obtained by rotating point T by angle pi/12 about point D. Let F be the incenter of triangle DXY. Line B is the perpendicular bisector of segment XY. Line V passes through point F parallel to line B. Point L is the centroid of triangle DXY. Point Z is the reflection of point T across line V. Find the measure of angle YZL, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let YM = 5. Point Z is obtained by rotating point Y by angle 11pi/12 about point M. Point G is obtained by rotating point M by angle pi/12 about point Y. Let Q be the reflection of point Y across point Z. Let R be the reflection of point Z across point Q. Let K be the midpoint of segment ZG. Find the circumradius of triangle ZRK. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let MN = 12. Let Z be the rotation of point M by angle 5pi/6 around point N. Let O be the midpoint of segment ZM. Let L be the circumcenter of triangle ONZ. Let V be the incenter of triangle ONZ. What is the distance between points L and V? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ON = 7. Point W is the reflection of point N across point O. Let K be the rotation of point O by angle pi/12 around point N. Line U is the perpendicular bisector of segment NW. Point M is the reflection of point N across segment WK. Point A is the reflection of point M across line U. Find the length of segment AM. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
QU = 12. Let E be the reflection of point Q across point U. Point J is obtained by rotating point E by angle 3pi/4 about point U. Let Z be the reflection of point Q across point J. Point N is the midpoint of segment ZJ. Point X is the reflection of point U across segment EZ. Find the inradius of triangle XNE. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
WF = 2. Let D be the reflection of point W across point F. Let O be the rotation of point F by angle pi/2 around point D. Point H is obtained by rotating point D by angle pi/4 about point F. Let A be the centroid of triangle OWF. Point U is the reflection of point O across segment WF. Let G be the incenter of triangle OWF. Point J is the reflection of point H across segment UA. Find the inradius of triangle WGJ. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
LP = 9. Point R is obtained by rotating point L by angle pi/4 about point P. Line X is the perpendicular bisector of segment RP. Let W be the midpoint of segment PL. Line N is the reflection of line X across point L. Let F be the midpoint of segment WR. Let O be the angle bisector of angle FLW. Let T be the reflection of line N across line O. Line H is the reflection of line T across line N. Point Q is the reflection of point L across line H. What is the circumradius of triangle RQW? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
XE = 9. Point S is obtained by rotating point E by angle 5pi/6 about point X. Let C be the reflection of point X across point S. Let N be the vector from point S to point X. Let H be point E translated by vector N. Line W is the angle bisector of angle HSC. Point I is the midpoint of segment HX. Point M is the midpoint of segment IX. Point U is the reflection of point M across line W. Let Q be the midpoint of segment UM. What is the measure of angle HIQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let SE = 5. Let W be the rotation of point E by angle pi/12 around point S. Let V be the rotation of point W by angle pi/12 around point S. Point T is obtained by rotating point W by angle pi/3 about point S. Let P be the reflection of point T across point E. Let J be the midpoint of segment VW. Find the inradius of triangle PTJ. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
OY = 9. Point W is obtained by rotating point O by angle pi/2 about point Y. Let P be the rotation of point Y by angle 7pi/12 around point O. Point V is the reflection of point O across point Y. Let U be the angle bisector of angle OPY. Point D is the reflection of point W across line U. What is the inradius of triangle WVD? Express your answer as a decimal number rounded to 2 decimal places.

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Let IF = 8. Point B is obtained by rotating point I by angle 5pi/12 about point F. Point D is the incenter of triangle IBF. Point Z is the midpoint of segment DB. Let R be the circumcenter of triangle IBF. Find the length of segment ZR. Express your answer as a decimal number rounded to 2 decimal places.

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Let YA = 7. Point K is obtained by rotating point Y by angle pi/6 about point A. Point T is the reflection of point K across segment AY. Let V be the centroid of triangle KTA. What is the distance between points Y and V? Express your answer as a decimal number rounded to 2 decimal places.

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WP = 2. Line T is the perpendicular bisector of segment WP. Point A lies on line T at distance 10 from point W. Point U is the midpoint of segment PW. Let M be the midpoint of segment UA. Point L is the orthocenter of triangle WAP. What is the distance between points M and L? Express your answer as a decimal number rounded to 2 decimal places.

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VY = 12. Point C is the midpoint of segment YV. Point E is obtained by rotating point V by angle pi/2 about point C. Point Q is obtained by rotating point Y by angle pi/12 about point C. Let D be the incircle of triangle QVE. Let G be the reflection of circle D through point Y. Point B is the inversion of point Y with respect to circle G. Find the distance between points B and C. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
CP = 6. Point A is obtained by rotating point C by angle 11pi/12 about point P. Let F be the perpendicular bisector of segment AC. Let L be the reflection of point A across point C. Point M is the reflection of point L across line F. What is the distance between points M and L? Express your answer as a decimal number rounded to 2 decimal places.

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KU = 5. Point A is obtained by rotating point U by angle 5pi/12 about point K. Point S is the reflection of point K across point U. Let T be the perpendicular bisector of segment AS. Let L be the circumcenter of triangle KAU. Let H be the line through point L parallel to line T. Point E is the reflection of point U across line H. Find the distance between points K and E. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let CJ = 2. Point H is obtained by rotating point C by angle pi/3 about point J. Point D is the reflection of point C across point J. Point W is the reflection of point H across segment CJ. Let S be the circumcenter of triangle HCJ. What is the measure of angle DWS, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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QA = 10. Let V be the reflection of point Q across point A. Point P is obtained by rotating point V by angle pi/2 about point Q. Let F be the midpoint of segment QP. Point E is the reflection of point A across point Q. Line T passes through point E parallel to segment QP. Point C is the reflection of point F across line T. Find the inradius of triangle EPC. Express your answer as a decimal number rounded to 2 decimal places.

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Let PZ = 1. Point G is the midpoint of segment ZP. Point Y is obtained by rotating point Z by angle 3pi/4 about point G. Let S be the midpoint of segment YZ. Let Q be the midpoint of segment GZ. Let H be the angle bisector of angle SGY. Let V be the midpoint of segment YG. Point A is the reflection of point Q across line H. Let L be the midpoint of segment AV. Find the distance between points V and L. Express your answer as a decimal number rounded to 2 decimal places.

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


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Let GA = 11. Let S be the vector from point G to point A. Let P be the rotation of point A by angle 11pi/12 around point G. Let Q be the midpoint of segment GP. Point T is the translation of point P by vector S. Circle B is the circumcircle of triangle TQA. Point H is the incenter of triangle TQA. Let K be the reflection of circle B through point A. Point U is the centroid of triangle TQA. Point W is the reflection of point H across segment TQ. Point C is the inversion of point G with respect to circle K. What is the measure of angle WUC, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let MU = 2. Let K be the rotation of point M by angle pi/3 around point U. Let P be the rotation of point K by angle 5pi/6 around point M. Let C be the perpendicular bisector of segment PK. Point F is the reflection of point U across line C. Let S be the midpoint of segment UM. Let H be the reflection of point S across point K. Find the inradius of triangle FHK. Express your answer as a decimal number rounded to 2 decimal places.

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


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PU = 7. Point H is obtained by rotating point P by angle 2pi/3 about point U. Let I be the midpoint of segment PU. Point Q is the midpoint of segment HP. Let W be the midpoint of segment HU. What is the circumradius of triangle IWQ? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let JI = 2. Point T is obtained by rotating point J by angle 2pi/3 about point I. Let L be the midpoint of segment TJ. Circle R is the circumcircle of triangle JIT. Point K is the inversion of point L with respect to circle R. Point U is the incenter of triangle JIT. Let Q be the midpoint of segment KJ. Find the measure of angle QLU, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


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Let HZ = 5. Line Y is the perpendicular bisector of segment HZ. Point C is the reflection of point H across point Z. Point K is on line Y with |KZ| = 5. Let Q be the reflection of point H across point C. Point J is the reflection of point Q across segment KH. Point N is the reflection of point Q across line Y. Find the measure of angle NJK, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


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Let KU = 10. Let J be the vector from point K to point U. Point M is obtained by rotating point K by angle pi/3 about point U. Point P is the translation of point U by vector J. Let I be the midpoint of segment UP. Point D is the reflection of point I across point P. Let V be the incenter of triangle MKP. Let G be the line through point V parallel to segment KP. Point B is the midpoint of segment PM. Let T be the reflection of point B across point P. Point F is the reflection of point T across line G. What is the measure of angle DVF, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


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EG = 3. Let L be the perpendicular bisector of segment GE. Point D is on line L with |DG| = 1. Line A is the angle bisector of angle EGD. Line U is the reflection of line L across line A. Line Z is the reflection of line A across point D. Point M is the reflection of point G across line Z. Segment EG and line U intersect at point X. Let I be the incenter of triangle XMD. What is the length of segment XI? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let VT = 7. Point J is obtained by rotating point V by angle 2pi/3 about point T. Let C be the incenter of triangle TVJ. Let I be the reflection of point T across point V. Let Z be the midpoint of segment IC. Find the distance between points Z and V. Express your answer as a decimal number rounded to 2 decimal places.

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


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PS = 4. Let J be the rotation of point S by angle pi/12 around point P. Let V be the midpoint of segment PJ. Let O be the midpoint of segment VJ. Line T is the perpendicular bisector of segment OS. Point A is the reflection of point V across line T. Point E is the midpoint of segment JA. Point Q is the reflection of point A across segment JS. Find the circumradius of triangle PQE. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NI = 3. Let V be the rotation of point N by angle 7pi/12 around point I. Point W is obtained by rotating point I by angle 2pi/3 about point N. Point F is the midpoint of segment WV. Point R is the reflection of point V across segment NI. Let Z be the line through points F and R. Line Z and segment VN intersect at point T. What is the distance between points T and R? Express your answer as a decimal number rounded to 2 decimal places.

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


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DN = 8. Let I be the midpoint of segment DN. Let P be the rotation of point D by angle pi/2 around point N. Point E is the reflection of point P across point N. Point M is obtained by rotating point P by angle 11pi/12 about point N. Point L is the midpoint of segment IM. Point X is the midpoint of segment ML. Let G be the midpoint of segment LD. What is the measure of angle EXG, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let LB = 8. Point J is obtained by rotating point L by angle 3pi/4 about point B. Let C be the circumcenter of triangle LJB. Point A is the reflection of point J across point L. Let M be the midpoint of segment BL. Find the measure of angle MCA, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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AY = 5. Let R be the rotation of point A by angle 11pi/12 around point Y. Point L is the reflection of point R across point A. Let M be the reflection of point Y across point L. Point V is the midpoint of segment AM. Let J be the centroid of triangle RAV. Let X be the incenter of triangle RAV. What is the measure of angle JMX, 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 TH = 8. Let Z be the rotation of point H by angle pi/12 around point T. Point F is the reflection of point T across point H. Circle R is the incircle of triangle FZT. Point I is the reflection of point H across segment ZT. Point A is the circumcenter of triangle FZT. Let N be the reflection of circle R through point A. Point U is the inversion of point I with respect to circle N. What is the length of segment UA? Express your answer as a decimal number rounded to 2 decimal places.

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Let UG = 12. Point Z is the reflection of point U across point G. UR = 12. Point B is the reflection of point R across point Z. Let W be the circumcenter of triangle UZB. Point S is the reflection of point W across segment UZ. What is the distance between points S and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


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US = 4. Point D is obtained by rotating point U by angle pi/3 about point S. Let H be the orthocenter of triangle DUS. Point I is the reflection of point D across segment US. Point L is the reflection of point I across segment DU. Find the length of segment LH. Express your answer as a decimal number rounded to 2 decimal places.

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HS = 2. Point M is obtained by rotating point S by angle 5pi/12 about point H. Let G be the line through point M parallel to segment SH. Point O is the reflection of point H across line G. Point D is the reflection of point M across segment SH. What is the circumradius of triangle DOS? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BN = 3. Point H is the reflection of point B across point N. Let T be the reflection of point H across point B. Line J is the angle bisector of angle HBT. Point M lies on line J at distance 5 from point N. Point Y is the reflection of point B across point H. Point P is obtained by rotating point Y by angle pi about point M. Line K passes through point N perpendicular to segment NT. Point R is the reflection of point P across line K. Let X be the incenter of triangle NPR. Find the length of segment XH. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let XY = 2. Line Q is the perpendicular bisector of segment XY. Point F is on line Q with |FX| = 11. Let G be the reflection of point F across point X. Let D be the line through point Y perpendicular to segment XY. Let H be the line through point G perpendicular to segment XY. Point W is the reflection of point F across line D. Point O is the reflection of point X across line H. Find the circumradius of triangle WFO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SR = 7. Point K is obtained by rotating point R by angle 5pi/6 about point S. Let L be the midpoint of segment KR. Point M is the midpoint of segment SK. Point U is the reflection of point L across point M. Find the circumradius of triangle UKR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MC = 12. Point T is the reflection of point C across point M. Let I be the rotation of point M by angle 3pi/4 around point C. Let F be the perpendicular bisector of segment MI. Let A be the centroid of triangle TMI. Point P is the midpoint of segment AC. Point J is the reflection of point P across line F. What is the distance between points T and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BR = 6. Let X be the rotation of point R by angle pi around point B. Point S is the reflection of point R across point X. Point J is the reflection of point X across point R. Point V is obtained by rotating point B by angle 3pi/4 about point J. Let D be the line through points B and V. Point G is the reflection of point S across line D. Let P be the midpoint of segment BR. Find the inradius of triangle PVG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZJ = 5. Point G is the reflection of point Z across point J. Line Y is the perpendicular bisector of segment ZG. Point T is the midpoint of segment GJ. Point X is on line Y with |XT| = 5. Let Q be the line through point Z parallel to line Y. Point O is the reflection of point T across line Q. Let V be the reflection of point O across point X. Point P is the midpoint of segment OG. What is the measure of angle VPT, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


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RJ = 1. Point K is obtained by rotating point J by angle pi/4 about point R. Line C is the perpendicular bisector of segment JR. Let M be the rotation of point J by angle 3pi/4 around point R. Let T be the midpoint of segment KM. Point O is the reflection of point T across line C. Point A is the midpoint of segment OJ. Line V passes through point M parallel to segment KR. Let E be the reflection of line V through point R. Point L is the reflection of point T across line E. What is the length of segment AL? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TE = 10. Point P is the midpoint of segment TE. Let I be the rotation of point E by angle 7pi/12 around point T. Let L be the midpoint of segment IP. Let N be the midpoint of segment EI. Point B is the reflection of point N across segment PE. Let R be the midpoint of segment BI. What is the distance between points R and L? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
IL = 5. Let A be the midpoint of segment IL. Line M is the perpendicular bisector of segment IL. Let F be the line through point L parallel to line M. Point S is the midpoint of segment IA. Let K be the rotation of point I by angle 5pi/12 around point S. Let W be the reflection of point L across point K. Point D is the reflection of point K across line F. Line X passes through points I and W. Point U is the reflection of point A across line X. Let R be the midpoint of segment UL. Find the circumradius of triangle IDR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CJ = 11. Point I is obtained by rotating point C by angle 3pi/4 about point J. Let V be the rotation of point C by angle 11pi/12 around point J. Point L is the reflection of point C across point V. Let K be the circumcenter of triangle IVL. Point H is the orthocenter of triangle IVL. What is the measure of angle HKL, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CG = 7. Point X is obtained by rotating point C by angle pi/6 about point G. Point E is the reflection of point G across point C. Let R be the rotation of point G by angle pi/2 around point X. Let U be the angle bisector of angle GXR. Point B is the reflection of point C across line U. Point T is the midpoint of segment GX. What is the circumradius of triangle TBE? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EN = 11. Point F is the reflection of point N across point E. Let T be the reflection of point F across point N. Line L is the perpendicular bisector of segment ET. Point O lies on line L at distance 4 from point N. Point G lies on line L at distance 6 from point F. Point P is the reflection of point G across point N. Point H is the reflection of point P across line L. Point R is the reflection of point F across segment OH. What is the distance between points R and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EM = 2. Point D is obtained by rotating point M by angle pi/6 about point E. Let A be the midpoint of segment ME. Point Z is the centroid of triangle DAE. Point U is the incenter of triangle DAE. What is the length of segment ZU? Express your answer as a decimal number rounded to 2 decimal places.

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


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AO = 4. Let E be the rotation of point O by angle pi/3 around point A. Point X is obtained by rotating point E by angle pi/4 about point O. Let Z be the midpoint of segment OX. Point Y is the reflection of point X across segment AZ. Point I is the incenter of triangle EAZ. What is the circumradius of triangle OIY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let WG = 3. Let Q be the vector from point G to point W. Point Y is the translation of point W by vector Q. Line M is the perpendicular bisector of segment WG. Point O is the midpoint of segment YW. Let J be the rotation of point G by angle 2pi/3 around point O. Let L be the line through point O parallel to line M. Point T is obtained by rotating point J by angle 3pi/4 about point Y. Line I is the angle bisector of angle GTW. Line V is the reflection of line I across point Y. Line U is the reflection of line M across line V. Lines L and U intersect at point X. Find the inradius of triangle YXT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SN = 2. Let P be the perpendicular bisector of segment SN. Point Y is obtained by rotating point S by angle pi/2 about point N. Point K is obtained by rotating point N by angle 5pi/6 about point Y. Point G is the reflection of point Y across line P. Point Q is the reflection of point K across segment SG. What is the circumradius of triangle QGY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HS = 2. Point T is obtained by rotating point H by angle pi/6 about point S. Line W passes through points S and T. Point C is the reflection of point H across line W. Point E is the reflection of point C across point T. Point R is the midpoint of segment HE. What is the circumradius of triangle SRH? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FI = 12. Let R be the rotation of point I by angle 2pi/3 around point F. Let P be the circumcenter of triangle RIF. Let H be the incenter of triangle RIF. Point G is the midpoint of segment IH. Let T be the midpoint of segment PF. Point Q is the midpoint of segment TP. Find the length of segment QG. Express your answer as a decimal number rounded to 2 decimal places.

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


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Let NG = 2. Let V be the rotation of point G by angle 5pi/6 around point N. Point Z is the reflection of point G across point N. Let H be the line through point N perpendicular to segment GN. Point T is the midpoint of segment VZ. Point Q is the reflection of point T across line H. Point E is the reflection of point Q across segment GN. Find the inradius of triangle EQG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WY = 9. Point A is the midpoint of segment YW. Point T is obtained by rotating point W by angle pi/12 about point A. Point N is the reflection of point Y across segment TA. Let I be the midpoint of segment NT. What is the distance between points T and I? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CS = 8. Let I be the rotation of point S by angle pi/6 around point C. Point K is the reflection of point I across point C. Let D be the angle bisector of angle CSK. Point G is the reflection of point K across line D. Let X be the reflection of line D through point K. Point V is the midpoint of segment GC. Let P be the midpoint of segment VS. Point Q is the reflection of point K across line X. What is the circumradius of triangle QPG? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RM = 2. Point C is obtained by rotating point R by angle pi/6 about point M. Line H is the perpendicular bisector of segment MR. Let T be the midpoint of segment RC. Let I be the line through point T parallel to line H. Point K is the reflection of point C across segment MR. Point L is the reflection of point K across line I. Point U is the reflection of point L across line H. Point W is the midpoint of segment KC. Find the measure of angle TWU, 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 DQ = 6. Line A is the perpendicular bisector of segment DQ. Let O be the rotation of point Q by angle 7pi/12 around point D. Let F be the reflection of point Q across point O. Let I be the reflection of point D across point Q. Point G is the reflection of point I across line A. What is the inradius of triangle FGQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BR = 4. Line M is the perpendicular bisector of segment BR. Point O is on line M with |OB| = 5. Line N passes through point R parallel to line M. Let S be the perpendicular bisector of segment OB. Let E be the reflection of line S across line N. Let A be the reflection of point O across point B. Point X is the orthocenter of triangle ORB. Point U is the reflection of point X across line E. Let P be the midpoint of segment OR. What is the measure of angle AUP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


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FJ = 7. Let I be the rotation of point F by angle 5pi/12 around point J. Line U passes through points I and F. Point W is obtained by rotating point I by angle pi/4 about point F. Let X be the line through point W perpendicular to line U. Point Y is the midpoint of segment FW. Point Z is the reflection of point I across line X. Find the circumradius of triangle ZYW. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CA = 8. Let G be the rotation of point A by angle pi/3 around point C. Point E is the midpoint of segment CA. Point H is the reflection of point E across point C. Let M be the reflection of point C across point H. Let I be the midpoint of segment GE. Find the measure of angle MHI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


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SN = 2. Let L be the rotation of point S by angle 2pi/3 around point N. Let D be the reflection of point N across point L. Let X be the rotation of point N by angle 5pi/12 around point L. Let A be the midpoint of segment LX. Line T is the angle bisector of angle XSN. Point E is the reflection of point D across line T. Point Y is the orthocenter of triangle AEX. What is the distance between points Y 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 ZD = 3. Let U be the midpoint of segment ZD. Let W be the rotation of point U by angle 11pi/12 around point D. Let I be the rotation of point Z by angle 5pi/12 around point W. Point F is the reflection of point Z across point D. Point Y is the midpoint of segment DF. Point K is the midpoint of segment DW. Point E is the reflection of point Y across point F. Point R is the midpoint of segment IK. What is the length of segment RF? Express your answer as a decimal number rounded to 2 decimal places.

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


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SP = 2. Let A be the rotation of point P by angle 3pi/4 around point S. Let E be the midpoint of segment PS. Let B be the incircle of triangle PAS. Let V be the inversion of point E in circle B. Let F be the midpoint of segment AE. What is the measure of angle FVP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
AB = 4. Point F is the midpoint of segment BA. Point L is obtained by rotating point A by angle 3pi/4 about point F. Point V is obtained by rotating point L by angle pi/6 about point A. Point J is obtained by rotating point B by angle 11pi/12 about point A. Let Y be the angle bisector of angle BVJ. Let R be the line through point A parallel to line Y. Point T is the reflection of point B across line R. What is the inradius of triangle JAT? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BE = 11. Point Y is the midpoint of segment EB. Let G be the rotation of point E by angle 11pi/12 around point B. Point W is the reflection of point B across point G. Let I be the angle bisector of angle BWE. Point D is the reflection of point E across line I. Point A is the reflection of point G across segment DE. What is the measure of angle AYD, 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 JF = 3. Let V be the rotation of point J by angle pi/3 around point F. Let P be the midpoint of segment JF. Point Q is the reflection of point V across point J. Point Y is the incenter of triangle VPJ. What is the circumradius of triangle YQF? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let FJ = 8. Let N be the rotation of point J by angle 5pi/6 around point F. Let S be the centroid of triangle NFJ. Let R be the rotation of point N by angle pi/12 around point F. Let W be the orthocenter of triangle NFJ. Let A be the midpoint of segment RF. Point O is obtained by rotating point W by angle pi/6 about point N. Let T be the rotation of point O by angle pi/6 around point J. Point Z is the reflection of point S across point O. Line X passes through point Z parallel to segment JN. Line P passes through point T perpendicular to line X. Line B is the perpendicular bisector of segment SA. Line I is the reflection of line P across line B. Lines I and X intersect at point Y. Let K be the midpoint of segment JN. Find the measure of angle YKA, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YQ = 6. Let X be the reflection of point Q across point Y. Let H be the rotation of point X by angle 11pi/12 around point Q. Point Z is obtained by rotating point H by angle 5pi/6 about point X. Let F be the midpoint of segment XH. Point N is the circumcenter of triangle HZF. Point R is the reflection of point X across segment ZF. Find the measure of angle RNQ, 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 MN = 10. Let G be the rotation of point N by angle 5pi/12 around point M. Point C is the circumcenter of triangle NMG. Let J be the circumcircle of triangle NMG. Let V be the reflection of point C across point G. Let U be the inversion of point V in circle J. Find the length of segment UC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QV = 4. Let C be the rotation of point V by angle 11pi/12 around point Q. Point B is obtained by rotating point C by angle pi/6 about point Q. Let Z be the reflection of point Q across point V. Let U be the midpoint of segment BV. Point M is the midpoint of segment BC. Point W is the reflection of point U across segment ZQ. What is the measure of angle UMW, 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 KJ = 5. Let P be the midpoint of segment JK. Let W be the rotation of point K by angle 11pi/12 around point P. Let L be the reflection of point P across point K. Let N be the reflection of point W across point K. Point U is the midpoint of segment PW. Let Z be the reflection of point U across point N. Find the length of segment LZ. Express your answer as a decimal number rounded to 2 decimal places.

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


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GQ = 3. Point F is obtained by rotating point G by angle 3pi/4 about point Q. Let P be the reflection of point Q across point G. Point H is the reflection of point G across point Q. Line Y is the perpendicular bisector of segment QF. Point M is obtained by rotating point F by angle 2pi/3 about point G. Let T be the reflection of point M across point G. Let Z be the incircle of triangle FPQ. Line I passes through point T perpendicular to line Y. Point R is the reflection of point F across segment PQ. Let L be the line through point G parallel to line I. Point J is the reflection of point R across line L. Let W be the inversion of point H in circle Z. 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}


[User]
MX = 3. Point H is the midpoint of segment XM. Let W be the rotation of point M by angle 5pi/12 around point H. Let A be the rotation of point X by angle pi/3 around point M. Point I is the reflection of point H across segment WA. Point O is the midpoint of segment MW. Let V be the incenter of triangle XWA. Point K is the centroid of triangle VOI. What is the distance between points H and K? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OZ = 11. Let D be the rotation of point Z by angle 2pi/3 around point O. Let U be the midpoint of segment ZO. Point V is the reflection of point O across segment UD. Point W is the reflection of point Z across segment UD. Find the circumradius of triangle VWO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AK = 2. Point N is obtained by rotating point A by angle 7pi/12 about point K. Point E is the reflection of point A across point N. Point Q is the midpoint of segment EN. Let O be the midpoint of segment QN. Point W is the incenter of triangle ENK. 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]
OV = 4. Let B be the line through points O and V. Line R passes through point V perpendicular to line B. Point X is the reflection of point O across line R. Let C be the reflection of point O across point X. Point L lies on line R at distance 8 from point V. Point Q is the reflection of point L across point X. Line D is the reflection of line B across point L. Point U is the reflection of point C across line D. Let I be the reflection of point V across point Q. What is the circumradius of triangle XIU? Express your answer as a decimal number rounded to 2 decimal places.

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FW = 12. Let P be the rotation of point F by angle pi/6 around point W. Let I be the angle bisector of angle PFW. Point J is the reflection of point W across line I. Let Q be the midpoint of segment WJ. What is the distance between points J and Q? Express your answer as a decimal number rounded to 2 decimal places.

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Let JW = 9. Point I is obtained by rotating point J by angle pi/3 about point W. Point Y is the reflection of point I across point J. Let T be the incenter of triangle WIY. Point H is the centroid of triangle WIY. What is the length of segment HT? Express your answer as a decimal number rounded to 2 decimal places.

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PJ = 8. V is constructed so that JV = 8. Point Q is the midpoint of segment PJ. Let B be the midpoint of segment VP. Point Y is the centroid of triangle VPQ. What is the length of segment YB? Express your answer as a decimal number rounded to 2 decimal places.

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Let GJ = 1. Point F is obtained by rotating point J by angle pi/6 about point G. Point M is the circumcenter of triangle GJF. Let P be the line through point M parallel to segment JF. Point R is the reflection of point J across line P. What is the length of segment MR? Express your answer as a decimal number rounded to 2 decimal places.

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Let GW = 4. Let S be the rotation of point W by angle pi/2 around point G. Let R be the perpendicular bisector of segment WG. Circle C is the incircle of triangle WGS. Let N be the reflection of circle C across line R. Point J is the midpoint of segment SW. Let O be the inversion of point W in circle N. Find the distance between points J and O. Express your answer as a decimal number rounded to 2 decimal places.

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WN = 8. Point C is obtained by rotating point N by angle pi/3 about point W. Point B is the reflection of point C across point N. Point X is obtained by rotating point N by angle 5pi/6 about point C. Point H is the reflection of point W across point N. Let Q be the reflection of point H across point C. Point L is the reflection of point W across segment NX. Let S be the reflection of point C across point B. What is the measure of angle LQS, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let IF = 9. Let T be the midpoint of segment IF. Point X is obtained by rotating point I by angle pi/6 about point F. Point H is the reflection of point F across point X. Line L is the angle bisector of angle FHT. Point O is the midpoint of segment TF. Point R is the reflection of point O across line L. What is the measure of angle HTR, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let RM = 12. Let N be the rotation of point R by angle 7pi/12 around point M. Point J is the midpoint of segment RM. Let K be the circumcenter of triangle MNJ. Let S be the reflection of point R across point K. Point F is the reflection of point N across point S. Find the distance between points F and R. Express your answer as a decimal number rounded to 2 decimal places.

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Let EA = 11. Let O be the rotation of point E by angle 7pi/12 around point A. Let R be the rotation of point E by angle 5pi/6 around point A. Point W is the incenter of triangle ARO. Point N is the reflection of point W across segment AR. Point C is the reflection of point W across point N. Let D be the midpoint of segment AR. What is the length of segment DC? Express your answer as a decimal number rounded to 2 decimal places.

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Let OH = 3. Let R be the rotation of point O by angle pi/2 around point H. Let W be the reflection of point R across point O. Point C is the reflection of point R across point W. Point T is the reflection of point R across segment HO. Point D is the reflection of point W across segment HO. Find the measure of angle TDC, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let UK = 6. Let D be the midpoint of segment UK. Let V be the rotation of point K by angle pi/6 around point D. Point X is the midpoint of segment VU. Let G be the line through points U and K. Point R is the reflection of point X across line G. Find the inradius of triangle DXR. Express your answer as a decimal number rounded to 2 decimal places.

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JB = 1. Let L be the rotation of point J by angle 3pi/4 around point B. Point T is the midpoint of segment LB. Let C be the circumcircle of triangle BJL. Let F be the circumcenter of triangle BJL. Let H be the inversion of point F in circle C. What is the length of segment HT? Express your answer as a decimal number rounded to 2 decimal places.

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WY = 1. Let V be the reflection of point W across point Y. Let X be the angle bisector of angle VYW. Line S is the perpendicular bisector of segment YW. Point L is obtained by rotating point Y by angle pi/4 about point W. Point F is the reflection of point L across point W. Point Q is obtained by rotating point L by angle pi/2 about point F. Let I be the midpoint of segment VL. Let M be the reflection of line X through point Q. Point A is on line M. Let P be the midpoint of segment WQ. Point B is the reflection of point P across line S. Find the measure of angle FBI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let BN = 1. Let L be the rotation of point B by angle 11pi/12 around point N. Point I is obtained by rotating point N by angle pi/2 about point B. Let J be the reflection of point B across point I. Let Q be the midpoint of segment LN. Point E is the reflection of point Q across segment JB. Let U be the incenter of triangle BLJ. What is the measure of angle JUE, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let LN = 6. Point D is obtained by rotating point L by angle pi/2 about point N. Let A be the midpoint of segment LD. Line Q passes through point L perpendicular to segment ND. Circle S is the circumcircle of triangle DAN. Point M is the reflection of point D across line Q. Point W is the circumcenter of triangle DAN. Point T is the reflection of point M across segment DA. Point B is the inversion of point L with respect to circle S. What is the measure of angle WTB, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let LC = 2. Let Q be the line through points L and C. Point A is the reflection of point L across point C. Line D is the perpendicular bisector of segment AC. Point R is on line D with |RC| = 9. Point T is the reflection of point R across line Q. Point F is the reflection of point A across point T. Point K is the reflection of point C across point F. What is the inradius of triangle KRL? Express your answer as a decimal number rounded to 2 decimal places.

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Let VR = 8. Point Q is obtained by rotating point V by angle 3pi/4 about point R. Let K be the midpoint of segment RQ. Let W be the orthocenter of triangle KVQ. Let P be the reflection of point R across point V. Find the distance between points P and W. Express your answer as a decimal number rounded to 2 decimal places.

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HZ = 2. Point I is obtained by rotating point H by angle pi/3 about point Z. Point P is the reflection of point I across point H. Line E is the perpendicular bisector of segment IP. Let J be the midpoint of segment PZ. Point W is the reflection of point Z across line E. Find the circumradius of triangle WZJ. Express your answer as a decimal number rounded to 2 decimal places.

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Let ZY = 6. Point A is obtained by rotating point Y by angle 3pi/4 about point Z. Line E is the perpendicular bisector of segment AZ. Point S is the reflection of point Z across point A. Point W is the reflection of point S across line E. Point N is the midpoint of segment WY. Let L be the reflection of point Y across point A. Find the measure of angle ZNL, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let BN = 3. Point J is the reflection of point B across point N. Point A is obtained by rotating point B by angle pi/6 about point N. Let M be the reflection of point N across point B. Let U be the orthocenter of triangle AMJ. Point P is the midpoint of segment AM. Find the measure of angle NUP, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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PG = 4. Let K be the rotation of point G by angle pi/4 around point P. Point F is obtained by rotating point P by angle pi/12 about point K. Let C be the orthocenter of triangle GFP. Point M is the incenter of triangle GFP. Point Q is the reflection of point M across segment FK. What is the length of segment QC? Express your answer as a decimal number rounded to 2 decimal places.

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Let HL = 4. Point R is obtained by rotating point H by angle 2pi/3 about point L. Point D is the reflection of point R across point L. Point K is the circumcenter of triangle LHR. Let S be the midpoint of segment KD. What is the distance between points S and K? Express your answer as a decimal number rounded to 2 decimal places.

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Let UK = 2. Let H be the rotation of point U by angle 5pi/6 around point K. Let P be the midpoint of segment KH. Let C be the circumcircle of triangle UHK. Let N be the reflection of circle C through point P. Let Z be the inversion of point P in circle N. Let R be the incenter of triangle UHK. Find the distance between points Z and R. Express your answer as a decimal number rounded to 2 decimal places.

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Let CA = 8. Line M is the perpendicular bisector of segment CA. Line W is the reflection of line M across point A. Point U lies on line M at distance 5 from point A. Point G is the reflection of point A across line W. Point T is the midpoint of segment AU. Let V be the orthocenter of triangle UAG. Find the circumradius of triangle VGT. Express your answer as a decimal number rounded to 2 decimal places.

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Let KD = 4. Let C be the rotation of point K by angle pi around point D. Line G is the angle bisector of angle CKD. Let A be the rotation of point C by angle pi/3 around point K. Let M be the rotation of point K by angle pi/2 around point C. Let F be the midpoint of segment MK. Point R is the midpoint of segment AC. Point Y is the reflection of point M across line G. What is the circumradius of triangle YRF? Express your answer as a decimal number rounded to 2 decimal places.

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UT = 7. Point E is the midpoint of segment TU. Let J be the rotation of point T by angle pi/4 around point E. Point R is obtained by rotating point J by angle pi about point U. Let V be the rotation of point J by angle pi/2 around point U. Point P is obtained by rotating point R by angle 3pi/4 about point T. Let K be the midpoint of segment VE. Point I is the incenter of triangle VJK. Let H be the reflection of point P across point I. Find the distance between points H and I. Express your answer as a decimal number rounded to 2 decimal places.

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Let NY = 10. Line T is the perpendicular bisector of segment NY. Point U lies on line T at distance 1 from point Y. Let L be the midpoint of segment NU. Line D passes through point N parallel to line T. Let E be the incenter of triangle YNU. Point R is the circumcenter of triangle YNU. Point X is the reflection of point E across line D. Find the measure of angle RXL, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let WC = 3. Let E be the midpoint of segment WC. Let D be the rotation of point W by angle pi/6 around point C. Point H is the reflection of point D across point W. Point A is the midpoint of segment DE. Let T be the incenter of triangle CHA. Find the distance between points H and T. Express your answer as a decimal number rounded to 2 decimal places.

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EI = 10. Point F is obtained by rotating point I by angle pi/2 about point E. Point T is the midpoint of segment FE. Point R is the incenter of triangle TFI. Let K be the orthocenter of triangle TFI. What is the distance between points K and R? Express your answer as a decimal number rounded to 2 decimal places.

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Let BH = 4. Line O is the perpendicular bisector of segment HB. Point C is on line O with |CB| = 8. Point F is the reflection of point H across point B. Point Y is the reflection of point F across line O. Point D is the centroid of triangle HCF. What is the length of segment DY? Express your answer as a decimal number rounded to 2 decimal places.

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CB = 11. Point H is the midpoint of segment CB. Point Y is the reflection of point B across point C. Let S be the rotation of point B by angle 3pi/4 around point Y. Line Q passes through point B perpendicular to segment SH. Line U passes through point C parallel to line Q. Point G is the intersection of segment SH and line U. Find the distance between points C and G. Express your answer as a decimal number rounded to 2 decimal places.

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Let MH = 8. Let B be the rotation of point H by angle pi/4 around point M. Line V passes through points B and H. Point G is obtained by rotating point B by angle 5pi/6 about point H. Point I is the reflection of point M across line V. Line K passes through point M parallel to line V. Point P is the midpoint of segment MB. Point X is the reflection of point G across line K. Find the measure of angle XIP, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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RI = 7. Point V is obtained by rotating point I by angle pi/4 about point R. Point W is the reflection of point V across point R. Let A be the perpendicular bisector of segment RV. Point O is the reflection of point W across line A. Let J be the circumcenter of triangle WVI. Let X be the angle bisector of angle JIW. Point Y is the reflection of point W across line X. Find the distance between points Y and O. Express your answer as a decimal number rounded to 2 decimal places.

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NI = 9. Let P be the reflection of point I across point N. Let O be the reflection of point P across point I. Let M be the midpoint of segment NO. Let B be the rotation of point M by angle 11pi/12 around point O. Line C passes through points B and O. Point H is the reflection of point N across line C. Point E is the reflection of point N across point I. What is the circumradius of triangle MEH? Express your answer as a decimal number rounded to 2 decimal places.

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Let RL = 7. Let U be the rotation of point R by angle pi/6 around point L. Let W be the rotation of point R by angle pi/2 around point U. Point P is the incenter of triangle WLU. Point H is the midpoint of segment PW. What is the circumradius of triangle HRW? Express your answer as a decimal number rounded to 2 decimal places.

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XN = 7. Let K be the rotation of point X by angle pi/3 around point N. Let J be the rotation of point X by angle pi/2 around point K. Point S is the reflection of point K across point J. Let Q be the reflection of point N across point J. Point M is the reflection of point N across point S. Find the circumradius of triangle MQX. Express your answer as a decimal number rounded to 2 decimal places.

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Let SX = 10. Point D is the midpoint of segment SX. Let W be the perpendicular bisector of segment XD. Let I be the line through point X parallel to line W. Let T be the midpoint of segment DS. Point Y is the reflection of point S across line I. Point B is obtained by rotating point X by angle 5pi/12 about point T. Point V is the midpoint of segment SB. Point J is the reflection of point T across segment VY. Find the inradius of triangle TJB. Express your answer as a decimal number rounded to 2 decimal places.

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ID = 1. Let S be the rotation of point I by angle 5pi/12 around point D. Let B be the circumcenter of triangle SDI. Point O is the reflection of point B across segment SD. Point R is the reflection of point B across segment IS. Find the measure of angle ROI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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RH = 3. Point K is obtained by rotating point R by angle 5pi/6 about point H. Let Y be the reflection of point K across point R. Let Q be the midpoint of segment RK. Let S be the midpoint of segment QY. Find the distance between points Q and S. Express your answer as a decimal number rounded to 2 decimal places.

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EC = 11. Let N be the perpendicular bisector of segment EC. Let D be the reflection of point C across point E. Point B is on line N with |BC| = 12. Point I is the reflection of point D across line N. Point S is the reflection of point B across segment DC. Point U is the reflection of point I across segment CB. Find the distance between points U and S. Express your answer as a decimal number rounded to 2 decimal places.

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Let QF = 5. Point U is obtained by rotating point F by angle 7pi/12 about point Q. Let Y be the centroid of triangle UFQ. Line X passes through point F parallel to segment QU. Let A be the midpoint of segment QU. Line I passes through point Y parallel to line X. Point S is the intersection of line I and segment FQ. Point L is the midpoint of segment US. What is the measure of angle LAQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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QH = 2. Point S is the reflection of point H across point Q. Let R be the rotation of point H by angle 5pi/12 around point S. Let W be the rotation of point S by angle 2pi/3 around point H. Let G be the midpoint of segment SR. Let M be the angle bisector of angle GWQ. Point K is the reflection of point Q across line M. Find the circumradius of triangle KHR. Express your answer as a decimal number rounded to 2 decimal places.

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LX = 3. Line N passes through points L and X. Point U is obtained by rotating point L by angle 3pi/4 about point X. Let G be the midpoint of segment XU. Point J is the reflection of point G across line N. Let R be the midpoint of segment JL. Point D is the orthocenter of triangle XJR. Point Q is the centroid of triangle XJR. What is the measure of angle QDG, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let MA = 12. Point Z is the midpoint of segment MA. Let K be the rotation of point A by angle 11pi/12 around point Z. Point S is the incenter of triangle MKZ. Point X is the midpoint of segment ZS. Point Y is the centroid of triangle MKZ. What is the distance between points Y and X? Express your answer as a decimal number rounded to 2 decimal places.

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Let HD = 11. Point F is obtained by rotating point H by angle pi/2 about point D. Line Z is the perpendicular bisector of segment FH. Point T is the circumcenter of triangle DFH. Let N be the reflection of point F across point T. Point K is the reflection of point N across line Z. Let U be the incircle of triangle DFH. Point X is the inversion of point H with respect to circle U. Find the length of segment XK. Express your answer as a decimal number rounded to 2 decimal places.

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FT = 4. Point E is the reflection of point F across point T. Point G is obtained by rotating point E by angle pi/2 about point F. Let C be the rotation of point G by angle 3pi/4 around point F. Point I is the reflection of point C across point F. Point N is the reflection of point I across point E. Point Q is the reflection of point I across segment GN. Point Y is the centroid of triangle CGN. What is the measure of angle QYF, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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RJ = 3. Point M is obtained by rotating point J by angle pi/12 about point R. Let N be the rotation of point M by angle 2pi/3 around point R. Let T be the midpoint of segment JR. Point Q is the reflection of point N across point T. Find the inradius of triangle JNQ. Express your answer as a decimal number rounded to 2 decimal places.

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RW = 10. Point M is obtained by rotating point W by angle pi/4 about point R. Point G is the midpoint of segment WM. Let K be the reflection of point G across point R. Let B be the incircle of triangle WKM. Point U is the inversion of point M with respect to circle B. Let L be the line through point M parallel to segment WK. Let H be the centroid of triangle WKM. Point N is the reflection of point U across line L. Point J is the orthocenter of triangle NKH. What is the distance between points G and J? Express your answer as a decimal number rounded to 2 decimal places.

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TH = 10. Point B is obtained by rotating point T by angle 3pi/4 about point H. Let Z be the perpendicular bisector of segment TB. Point O is obtained by rotating point T by angle pi about point B. Circle V is the circumcircle of triangle OHB. Let K be the reflection of circle V across line Z. Point L is the inversion of point B with respect to circle K. Find the length of segment LT. Express your answer as a decimal number rounded to 2 decimal places.

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Let R be the midpoint of segment Z92L. Point B is the orthocenter of triangle L. Let F be the midpoint of segment UR. Let Z be the reflection of point L across point F. Point G is the midpoint of segment BF. What is the length of segment ZG? Express your answer as a decimal number rounded to 2 decimal places.

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YH = 11. Point D is the reflection of point Y across point H. Point K is obtained by rotating point H by angle pi/2 about point D. Line V passes through points D and H. Let N be the line through points K and H. Line B is the reflection of line N across line V. Point X is the midpoint of segment DH. Point C is the reflection of point Y across line B. What is the measure of angle KCX, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let VK = 8. Let F be the rotation of point K by angle 5pi/6 around point V. Let Q be the midpoint of segment FK. Point N is the midpoint of segment QK. Let J be the midpoint of segment NK. Point O is the centroid of triangle QVN. Find the distance between points J and O. Express your answer as a decimal number rounded to 2 decimal places.

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HC = 12. Point N is obtained by rotating point C by angle 3pi/4 about point H. Point M is the midpoint of segment CN. Point B is the incenter of triangle NHC. Let T be the midpoint of segment BM. Point P lies on segment NH. What is the measure of angle PHT, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let UK = 3. Let V be the rotation of point K by angle pi/3 around point U. Point G is the midpoint of segment UK. Point B is the reflection of point G across segment VK. Find the inradius of triangle KGB. Express your answer as a decimal number rounded to 2 decimal places.

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GP = 6. Point D is obtained by rotating point G by angle 11pi/12 about point P. Let A be the midpoint of segment DP. Let N be the reflection of point G across point P. Let T be the midpoint of segment NA. Point V is the midpoint of segment PA. What is the measure of angle VGT, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let UL = 10. Let B be the reflection of point U across point L. Let N be the rotation of point B by angle 5pi/6 around point U. Line O passes through points U and N. Let UT = 4. Point S is the reflection of point B across line O. Let E be the perpendicular bisector of segment BL. Point F is the reflection of point S across line E. What is the length of segment BF? Express your answer as a decimal number rounded to 2 decimal places.

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TK = 2. Point N is the midpoint of segment KT. Let P be the rotation of point K by angle 2pi/3 around point N. Point Z is the midpoint of segment PN. Point O is the reflection of point N across segment PT. Point D is the circumcenter of triangle ZPT. Find the measure of angle DPO, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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JP = 5. Point N is the reflection of point P across point J. Let U be the rotation of point P by angle pi/6 around point J. Line D is the angle bisector of angle UPJ. Line O is the angle bisector of angle UJP. Let A be the midpoint of segment JN. Let Y be the midpoint of segment JU. Line W passes through point A perpendicular to line O. Let I be the line through point N parallel to line W. Line L passes through point J parallel to line D. Point E is the intersection of lines L and I. Find the measure of angle EUY, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let UL = 9. Point M is the midpoint of segment UL. Let E be the angle bisector of angle UML. Point H is obtained by rotating point L by angle pi/12 about point U. Point P is obtained by rotating point M by angle 3pi/4 about point H. Line I passes through point L parallel to line E. Point G is the reflection of point P across line I. Let V be the midpoint of segment LG. Find the measure of angle MVU, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let DL = 3. Let K be the rotation of point L by angle pi/4 around point D. Point P is the midpoint of segment LK. Let B be the reflection of point P across point K. Let A be the circumcenter of triangle DKL. Let I be the midpoint of segment BA. Find the distance between points D and I. Express your answer as a decimal number rounded to 2 decimal places.

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TX = 3. Line N passes through points X and T. Point C is obtained by rotating point T by angle 5pi/12 about point X. Let G be the reflection of point X across point C. Let Y be the line through point C perpendicular to line N. Point O is on line Y. Let V be the midpoint of segment TG. Point W is the reflection of point C across point V. Point A is the reflection of point C across segment TG. What is the length of segment WA? Express your answer as a decimal number rounded to 2 decimal places.

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MI = 2. Let J be the rotation of point I by angle pi/6 around point M. Point X is obtained by rotating point I by angle pi/12 about point M. Point B is the midpoint of segment MJ. Point E is the reflection of point X across segment IM. Point A is the reflection of point B across segment IM. Point L is the midpoint of segment MX. What is the inradius of triangle ELA? Express your answer as a decimal number rounded to 2 decimal places.

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Let JP = 6. Let D be the rotation of point P by angle pi/4 around point J. M is placed 1 units away from D. Let C be the rotation of point M by angle pi/6 around point P. Point S is the reflection of point C across segment PM. Point U is the reflection of point P across point S. Point R is the midpoint of segment CS. What is the measure of angle RSU, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let UZ = 1. Let E be the reflection of point Z across point U. Let K be the rotation of point Z by angle pi/2 around point E. Point R is the circumcenter of triangle EKU. Point L is the orthocenter of triangle KRZ. Point X is the centroid of triangle KRZ. What is the measure of angle KXL, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
RO = 11. Let C be the line through points R and O. Point Y is obtained by rotating point O by angle 7pi/12 about point R. Let U be the perpendicular bisector of segment YR. Point L is the reflection of point Y across line C. Let I be the rotation of point Y by angle pi around point R. Let G be the rotation of point I by angle 2pi/3 around point R. Let Q be the reflection of line U through point O. Let A be the circumcircle of triangle YGL. Let V be the reflection of circle A across line Q. Point K is the inversion of point Y with respect to circle V. What is the measure of angle KYL, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
LK = 8. Point V is the reflection of point L across point K. Point D is obtained by rotating point K by angle pi/12 about point V. Point O is the midpoint of segment KD. Let M be the line through point O parallel to segment LK. Point Z is the reflection of point L across line M. Find the circumradius of triangle OVZ. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
SB = 9. Let W be the rotation of point S by angle 7pi/12 around point B. Let C be the rotation of point W by angle pi/6 around point B. Point K is the midpoint of segment WS. Line N is the perpendicular bisector of segment KC. Point D is the reflection of point W across line N. Line I is the angle bisector of angle CDS. Point T is obtained by rotating point D by angle pi about point W. Line G is the reflection of line I across point C. Line Z passes through point T perpendicular to line I. Point E is the reflection of point C across line Z. Point P is the reflection of point E across line G. What is the measure of angle PEK, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let LT = 1. Let R be the vector from point L to point T. Point E is obtained by rotating point T by angle 5pi/12 about point L. Point K is the translation of point T by vector R. Line F is the perpendicular bisector of segment LK. Let C be the midpoint of segment LE. Let O be the midpoint of segment LC. Point I is the reflection of point O across line F. What is the inradius of triangle KTI? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let LG = 3. Point P is the reflection of point G across point L. Point J is obtained by rotating point P by angle 5pi/12 about point G. Point V is obtained by rotating point J by angle pi/6 about point P. Let Y be the perpendicular bisector of segment PV. Point O is the orthocenter of triangle PVG. Point Q is the reflection of point G across line Y. Find the distance between points O and Q. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ZU = 9. Let H be the reflection of point Z across point U. Point M is obtained by rotating point H by angle pi/4 about point Z. Point V is the centroid of triangle HMZ. Point Q is the midpoint of segment ZM. Point S is the midpoint of segment UV. Point Y is the reflection of point S across segment MZ. Find the distance between points Y and Q. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ZG = 11. Point Q is the reflection of point Z across point G. Point J is obtained by rotating point G by angle 7pi/12 about point Z. Let F be the rotation of point Q by angle 11pi/12 around point G. Point X is the orthocenter of triangle FJG. Point W is the incenter of triangle FJG. Point D is the centroid of triangle FJG. Find the inradius of triangle WXD. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let HP = 12. Let I be the rotation of point H by angle pi/12 around point P. Point C is the midpoint of segment IP. Let Z be the midpoint of segment IC. Point T is the circumcenter of triangle ICH. What is the measure of angle TCZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let HS = 9. Line Z passes through points H and S. Point J is obtained by rotating point S by angle 11pi/12 about point H. Point G is the reflection of point J across point S. Point Y is the incenter of triangle GHJ. Let I be the reflection of line Z through point Y. Point C is the reflection of point J across line I. What is the distance between points G and C? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let WO = 3. Point L is obtained by rotating point W by angle 11pi/12 about point O. Point S is the midpoint of segment OW. Let B be the incenter of triangle LOW. Point X is the reflection of point B across segment WL. Point U is the midpoint of segment SX. What is the length of segment XU? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let BC = 9. Let L be the rotation of point C by angle 5pi/6 around point B. Let Q be the midpoint of segment CL. Let S be the midpoint of segment QL. Point Z is the reflection of point Q across segment BL. Find the measure of angle SBZ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
UZ = 2. Let I be the reflection of point Z across point U. Let J be the vector from point Z to point I. Let K be point I translated by vector J. Point D is obtained by rotating point U by angle 5pi/12 about point Z. Let F be the vector from point U to point I. Point R is the translation of point K by vector J. Let E be point D translated by vector F. Let G be the rotation of point R by angle pi around point E. Point M is obtained by rotating point G by angle pi/3 about point R. Point X is the midpoint of segment DZ. Point C is the midpoint of segment MX. Point B is the midpoint of segment KD. Find the inradius of triangle CBU. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let CH = 12. Let W be the rotation of point C by angle 5pi/12 around point H. Line N passes through point H parallel to segment CW. Line U is the reflection of line N across point W. Point E is the midpoint of segment HC. Point T is the reflection of point W across line U. Point L is the reflection of point E across line N. What is the measure of angle TEL, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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