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OQ = 12. Point E is obtained by rotating point Q by angle 5pi/6 about point O. Point I is the midpoint of segment QO. Let X be the line through point E perpendicular to segment EO. Point J is the reflection of point Q across line X. What is the circumradius of triangle IEJ? 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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GC = 6. Let Q be the midpoint of segment GC. Let B be the rotation of point Q by angle pi/4 around point G. Line X is the perpendicular bisector of segment BG. Let D be the midpoint of segment BQ. Line R passes through point G parallel to line X. Point E is the midpoint of segment DB. Point W is the reflection of point Q across line R. Find the length of segment EW. 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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ID = 11. Point R is the reflection of point D across point I. Let F be the rotation of point D by angle pi/4 around point I. Point Z is the reflection of point I across point F. Let J be the centroid of triangle RFZ. Point H is the reflection of point R across segment FZ. Find the length of segment JH. Express your answer as a decimal number rounded to 2 decimal places.

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


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IS = 3. Point F is obtained by rotating point I by angle 5pi/12 about point S. B is constructed so that FB = 3. Point V is the reflection of point S across segment IF. Let Q be the midpoint of segment IS. Let Y be the line through points B and V. Point K is the reflection of point Q across line Y. What is the distance between points K 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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IU = 6. Let O be the rotation of point U by angle pi/6 around point I. Point H is the midpoint of segment OU. Point M is the centroid of triangle OIH. Point S is the incenter of triangle OIH. What is the distance between points M and S? 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 TC = 3. Point E is obtained by rotating point T by angle pi/2 about point C. Point J is the reflection of point E across point C. Let H be the circumcenter of triangle TJE. Let L be the incenter of triangle TJE. Point A is the reflection of point L across point H. Find the distance between points T and A. 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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HE = 1. Let P be the rotation of point H by angle 5pi/12 around point E. Point I is the reflection of point E across point P. Point B is the midpoint of segment PH. Point X is the reflection of point I across point B. What is the inradius of triangle PBX? 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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AR = 5. Point C is obtained by rotating point A by angle 5pi/12 about point R. Let H be the perpendicular bisector of segment CR. Point G is the midpoint of segment AC. Point J is the reflection of point A across line H. Let V be the midpoint of segment JG. Point Y is the reflection of point V across segment AC. Find the circumradius of triangle RYC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AL = 3. Let R be the rotation of point L by angle pi/4 around point A. Let Q be the circumcenter of triangle ALR. Line Z passes through point Q parallel to segment AL. Segment LR and line Z intersect at point Y. Let F be the midpoint of segment LR. Find the distance between points Y and F. 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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BQ = 7. Point H is the midpoint of segment BQ. Point V is obtained by rotating point B by angle pi/6 about point Q. Line E passes through point H perpendicular to segment HB. Point D is the reflection of point V across segment HB. Point X is the reflection of point D across line E. Find the inradius of triangle BXV. 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 DR = 6. Let J be the rotation of point R by angle pi/3 around point D. Let N be the incenter of triangle DRJ. Let Z be the midpoint of segment NR. Let M be the midpoint of segment RJ. What is the measure of angle ZMN, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BR = 8. Let X be the reflection of point R across point B. Let Y be the rotation of point B by angle 7pi/12 around point X. Let F be the reflection of point B across point X. Circle A is the circumcircle of triangle RBY. Point D is the reflection of point Y across segment RB. Let V be the inversion of point F in circle A. Point P is the reflection of point V across point X. Point H is the circumcenter of triangle RBY. Find the measure of angle DPH, 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 UB = 10. Point D is obtained by rotating point U by angle 11pi/12 about point B. Let K be the angle bisector of angle BUD. Line W passes through point U perpendicular to line K. Point E is the reflection of point B across line W. Find the inradius of triangle EDB. 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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UH = 9. Point J is obtained by rotating point H by angle 3pi/4 about point U. Let F be the angle bisector of angle UHJ. Point M is the reflection of point J across line F. Point N is the midpoint of segment MH. Let P be the reflection of point H across point U. Point K is the centroid of triangle JUN. What is the distance between points K and P? 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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CW = 12. Point O is obtained by rotating point W by angle 5pi/6 about point C. Point T is the midpoint of segment CO. Point K is the midpoint of segment CW. Point X is the reflection of point C across point W. Find the circumradius of triangle TXK. 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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OD = 1. Let S be the midpoint of segment OD. Point K is obtained by rotating point D by angle pi/6 about point S. Let I be the midpoint of segment KO. Line F passes through points O and S. Let V be the reflection of point D across point I. Point B is the reflection of point V across line F. Find the inradius of triangle OBV. 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 QH = 4. Let O be the rotation of point H by angle 5pi/6 around point Q. Let J be the midpoint of segment OQ. Line V is the angle bisector of angle OJH. Point L is the midpoint of segment QH. Point E is the reflection of point L across line V. Find the measure of angle LEJ, 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 SF = 7. Line P passes through point S perpendicular to segment SF. Let X be the rotation of point S by angle 5pi/6 around point F. Let D be the midpoint of segment XS. Point O is the reflection of point D across line P. Let L be the centroid of triangle SDO. What is the distance between points L and O? 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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IV = 11. Point M is obtained by rotating point I by angle 3pi/4 about point V. Let S be the reflection of point M across point V. Point F is the midpoint of segment MV. Line Y is the angle bisector of angle SIV. Let H be the rotation of point F by angle pi around point I. Line W is the perpendicular bisector of segment MV. Let Q be the reflection of line Y through point H. Line U passes through point V parallel to line W. Lines Q and U intersect at point K. Find the measure of angle HKI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let RO = 5. Let Z be the perpendicular bisector of segment RO. Point T lies on line Z at distance 8 from point R. Point E is obtained by rotating point R by angle 7pi/12 about point T. Let V be the orthocenter of triangle OER. Point U is the midpoint of segment ER. Let F be the midpoint of segment VO. Find the distance between points U and F. 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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HE = 8. Point V is obtained by rotating point E by angle pi/3 about point H. Point I is the midpoint of segment EV. Point P is the centroid of triangle VHE. Let B be the midpoint of segment EP. Find the length of segment BI. Express your answer as a decimal number rounded to 2 decimal places.

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PA = 2. Point S is obtained by rotating point P by angle pi/6 about point A. Line L is the perpendicular bisector of segment SA. Let C be the midpoint of segment AS. Point W is the reflection of point P across line L. Let I be the reflection of point W across point S. What is the circumradius of triangle CSI? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ID = 6. Let N be the rotation of point I by angle 11pi/12 around point D. Line K is the perpendicular bisector of segment ID. Point A is obtained by rotating point N by angle pi about point I. Let U be the line through point A parallel to line K. Let Q be the rotation of point N by angle 11pi/12 around point A. Line J is the reflection of line K across point I. Point Y is the intersection of segment QD and line J. Point V is the reflection of point Y across line U. Find the measure of angle AVI, 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 AB = 11. Let D be the rotation of point B by angle pi/12 around point A. Point S is the reflection of point A across point D. Let F be the orthocenter of triangle BAS. Circle J is the circumcircle of triangle BAS. Point Q is the inversion of point F with respect to circle J. What is the distance between points Q and F? Express your answer as a decimal number rounded to 2 decimal places.

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


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ZK = 10. Point H is obtained by rotating point Z by angle 5pi/12 about point K. Let W be the midpoint of segment ZH. Let F be the midpoint of segment ZK. Point C is the reflection of point W across point Z. Line G is the perpendicular bisector of segment KH. Point Q is the reflection of point W across line G. Point X is the midpoint of segment FC. Find the distance between points Q and X. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XD = 7. Let Q be the midpoint of segment DX. Point O is obtained by rotating point Q by angle 2pi/3 about point X. Point Z is the midpoint of segment DQ. Point A is the reflection of point Q across point O. Let G be the midpoint of segment QZ. Point Y is the circumcenter of triangle AZQ. What is the length of segment GY? 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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GF = 2. Let R be the rotation of point G by angle 11pi/12 around point F. Circle E is the incircle of triangle GRF. Let U be the midpoint of segment RF. Point P is the inversion of point U with respect to circle E. What is the length of segment PR? 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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HS = 8. Let Q be the rotation of point H by angle pi/6 around point S. Point B is the midpoint of segment HS. Point X is the midpoint of segment QS. Point P is the midpoint of segment XQ. Find the measure of angle BPH, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KX = 7. Let N be the reflection of point X across point K. Point L is obtained by rotating point K by angle 3pi/4 about point N. Let H be the midpoint of segment KL. Let R be the angle bisector of angle HLX. Point V is the reflection of point K across line R. What is the inradius of triangle HVN? 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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YD = 11. Let V be the rotation of point Y by angle pi/12 around point D. Point N is the midpoint of segment VD. Let P be the orthocenter of triangle VNY. Let H be the centroid of triangle VNY. What is the measure of angle NHP, 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 UI = 3. Point V is obtained by rotating point U by angle 7pi/12 about point I. Point A is obtained by rotating point I by angle pi/2 about point U. Let X be the circumcircle of triangle IAV. Let F be the incenter of triangle IAV. Let Q be the inversion of point F in circle X. What is the length of segment AQ? 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 AE = 6. Point X is obtained by rotating point E by angle pi/2 about point A. Let V be the rotation of point X by angle 3pi/4 around point A. Point W is the reflection of point V across point E. Point C is the reflection of point V across point A. Let R be the midpoint of segment XV. Find the circumradius of triangle WCR. 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 WD = 3. Line G passes through points D and W. Point T is obtained by rotating point D by angle pi/4 about point W. Let N be the rotation of point D by angle 2pi/3 around point T. Let M be the perpendicular bisector of segment TN. Let K be the reflection of line M across line G. Let B be the reflection of line K across line M. Point S is the reflection of point D across line B. Find the circumradius of triangle NST. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZQ = 3. Point E is obtained by rotating point Q by angle 5pi/12 about point Z. Point I is obtained by rotating point Q by angle 3pi/4 about point E. Let P be the reflection of point E across point Q. Let G be the centroid of triangle EZI. What is the circumradius of triangle PIG? 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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UW = 12. Point D is the midpoint of segment WU. Point Y is the reflection of point D across point U. Point S is obtained by rotating point D by angle pi/6 about point Y. Let G be the reflection of point W across point S. Point T is obtained by rotating point U by angle pi/4 about point W. Let X be the circumcenter of triangle GTY. Let V be the incenter of triangle DGX. Let M be the circumcenter of triangle DGX. What is the distance between points M and V? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SL = 4. Let Y be the rotation of point S by angle 7pi/12 around point L. Point W is obtained by rotating point Y by angle 11pi/12 about point S. Let Z be the midpoint of segment YS. Let X be the reflection of point W across point Z. Point P is the reflection of point W across point Y. What is the length of segment PX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let ZQ = 6. Let V be the rotation of point Z by angle 7pi/12 around point Q. Point Y is the centroid of triangle QZV. Point C is the midpoint of segment ZY. Point F is the reflection of point Y across segment QZ. Let A be the midpoint of segment VF. Find the distance between points A and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let EA = 5. Let Z be the midpoint of segment EA. Let P be the reflection of point Z across point E. Let H be the rotation of point A by angle 3pi/4 around point P. Let S be the reflection of point H across point P. Point N is the reflection of point Z across segment HS. Line Q is the perpendicular bisector of segment ZH. Point I is the centroid of triangle HSZ. Let W be the midpoint of segment IN. Point R is the reflection of point N across line Q. Find the measure of angle PWR, 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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DL = 4. Point M is obtained by rotating point L by angle 2pi/3 about point D. Let G be the midpoint of segment DL. Point J is the midpoint of segment DM. Point Z is the centroid of triangle LJG. Find the length of segment DZ. 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 LD = 6. Let O be the line through points L and D. Let R be the rotation of point D by angle 3pi/4 around point L. Line E is the perpendicular bisector of segment RL. Let X be the reflection of line E through point L. Lines X and O intersect at point C. Let K be the midpoint of segment CD. Let V be the midpoint of segment KR. Point W lies on line E. Point Q is the reflection of point C across line E. Let A be the rotation of point C by angle pi/2 around point Q. Point H is the centroid of triangle AWD. Point T is the reflection of point V across segment DQ. What is the length of segment CT? Express your answer as a decimal number rounded to 2 decimal places.

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Let QS = 7. Point D is obtained by rotating point Q by angle pi/2 about point S. Circle T is the circumcircle of triangle DSQ. Let I be the angle bisector of angle SDQ. Point R is the reflection of point Q across line I. Let F be the incenter of triangle DSQ. Let Z be the inversion of point F in circle T. Find the length of segment ZR. 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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HC = 3. Point B is obtained by rotating point C by angle 5pi/12 about point H. Point U is obtained by rotating point H by angle pi/12 about point B. Line F passes through point H parallel to segment UC. Point O is the reflection of point C across line F. Point P is the reflection of point B across line F. Find the measure of angle POH, 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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IK = 5. Let W be the midpoint of segment IK. Point R is the midpoint of segment WI. Point E is obtained by rotating point R by angle 5pi/6 about point I. Point Z is the reflection of point E across point W. Circle D is the incircle of triangle ZKE. Let G be the perpendicular bisector of segment KE. Let J be the line through point R parallel to line G. Point Y is the reflection of point W across line G. Point H is the intersection of line J and segment KE. Let O be the inversion of point H in circle D. Find the measure of angle YOK, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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LX = 4. Let N be the rotation of point X by angle 5pi/12 around point L. Point T is the reflection of point N across point L. Point R is obtained by rotating point T by angle 5pi/6 about point L. Point Y is the midpoint of segment TX. Let K be the angle bisector of angle NTY. Point D is the reflection of point X across line K. Find the circumradius of triangle XRD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let GJ = 7. Point C is the reflection of point J across point G. Let N be the rotation of point G by angle pi/3 around point J. Let W be the reflection of point N across point J. Let F be the angle bisector of angle JWC. Line P is the perpendicular bisector of segment JG. Let L be the incircle of triangle NJG. Point V is the inversion of point J with respect to circle L. Line I is the reflection of line P across line F. Segment JG and line I intersect at point U. Find the length of segment UV. Express your answer as a decimal number rounded to 2 decimal places.

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Let VF = 5. Line G passes through points V and F. Line D passes through point V perpendicular to line G. Point S lies on line D at distance 2 from point F. Point A is the midpoint of segment SF. Line R is the angle bisector of angle SVA. Point U is the midpoint of segment SV. Point I is the reflection of point A across line R. Line W is the perpendicular bisector of segment FU. Point N is the reflection of point I across line W. Find the measure of angle INF, 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 WE = 3. Let F be the rotation of point W by angle 2pi/3 around point E. Let A be the reflection of point F across point W. Let M be the reflection of point F across point A. Point I is the reflection of point F across segment WE. Find the length of segment IM. Express your answer as a decimal number rounded to 2 decimal places.

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Let QT = 12. Point O is the reflection of point Q across point T. Line S is the angle bisector of angle OTQ. Point C is the reflection of point T across point O. Point U lies on line S at distance 6 from point C. Line N is the angle bisector of angle OQU. Line I is the reflection of line S across line N. Point P is the reflection of point C across line I. Find the inradius of triangle TQP. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RC = 8. Let RW = 12. Point J is the reflection of point W across point C. Let Z be the midpoint of segment JR. Let S be the centroid of triangle RCJ. 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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LQ = 4. Point U is the midpoint of segment LQ. Point X is obtained by rotating point Q by angle 2pi/3 about point L. Point W is obtained by rotating point U by angle pi/4 about point L. Let O be the reflection of point W across point X. Point Z is the reflection of point L across point O. Point R is the reflection of point X across point L. Find the inradius of triangle ZOR. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FE = 10. Point A is obtained by rotating point E by angle 11pi/12 about point F. Let U be the midpoint of segment EF. Let L be the reflection of point A across point F. Let T be the incenter of triangle EAF. What is the measure of angle TUL, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let LY = 4. Point J is obtained by rotating point L by angle pi about point Y. Let M be the line through points J and L. Let O be the rotation of point J by angle 5pi/12 around point L. Let H be the rotation of point J by angle pi/12 around point Y. Line C passes through point O perpendicular to line M. Point Q is the reflection of point J across line C. Line G is the perpendicular bisector of segment OY. Point D is the reflection of point J across line G. What is the measure of angle QHD, 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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CE = 6. Line U is the perpendicular bisector of segment EC. Let Q be the reflection of line U through point E. Point N is on line Q with |NE| = 12. Point H is the reflection of point E across line Q. Let V be the midpoint of segment HN. Point P is the circumcenter of triangle CNV. What is the length of segment PC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let CO = 7. Let U be the reflection of point O across point C. Let G be the rotation of point C by angle 3pi/4 around point O. Let B be the perpendicular bisector of segment OC. Line D is the reflection of line B across point C. Let S be the circumcenter of triangle UCG. Line D and segment GU intersect at point L. Point I is the midpoint of segment GL. What is the length of segment SI? Express your answer as a decimal number rounded to 2 decimal places.

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


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LT = 9. Let Y be the rotation of point L by angle 3pi/4 around point T. Let F be the midpoint of segment YT. Let D be the midpoint of segment FL. Let H be the orthocenter of triangle LYF. Let P be the reflection of point H across point D. What is the measure of angle PHL, 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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PZ = 7. Point I is obtained by rotating point Z by angle pi/2 about point P. Let O be the line through point I perpendicular to segment PI. Let B be the incircle of triangle PIZ. Point T is the reflection of point Z across line O. Let W be the inversion of point T in circle B. Let U be the midpoint of segment IZ. What is the distance between points U and W? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
SB = 10. Let Q be the reflection of point B across point S. Let X be the rotation of point B by angle 7pi/12 around point Q. Let J be the midpoint of segment SQ. Point R is obtained by rotating point S by angle 5pi/12 about point Q. Let D be the midpoint of segment JX. Let H be the circumcenter of triangle RQX. Point P is the reflection of point D across point Q. What is the distance between points P and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PC = 9. Let Y be the midpoint of segment PC. Let Z be the rotation of point Y by angle 7pi/12 around point P. Let H be the midpoint of segment YZ. Let M be the midpoint of segment CZ. Let G be the line through point P perpendicular to segment PC. Point D is the reflection of point P across point M. Point B is the reflection of point D across line G. What is the measure of angle YHB, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let MY = 4. Point W is obtained by rotating point Y by angle pi/3 about point M. Line U is the perpendicular bisector of segment WY. Line X is the reflection of line U across point W. Point R is the reflection of point W across line X. What is the inradius of triangle RMW? 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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YX = 1. Let D be the rotation of point X by angle 5pi/12 around point Y. Point W is the midpoint of segment YD. Point E is obtained by rotating point X by angle 11pi/12 about point Y. Point I is the midpoint of segment XW. Line F passes through points E and I. Point K is the reflection of point W across line F. Let R be the midpoint of segment XD. What is the circumradius of triangle WKR? 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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YR = 7. Let V be the rotation of point Y by angle 3pi/4 around point R. Let W be the rotation of point R by angle 2pi/3 around point Y. Point H is the circumcenter of triangle RVW. Point J is the reflection of point Y across segment VW. Let F be the centroid of triangle HJV. What is the distance between points F and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NA = 4. Point X is the midpoint of segment AN. Let G be the rotation of point A by angle 7pi/12 around point X. Point C is the orthocenter of triangle GXN. Point H is the centroid of triangle GXN. Find the circumradius of triangle NHC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VM = 5. Let L be the midpoint of segment VM. Let I be the rotation of point M by angle 5pi/6 around point L. Let D be the circumcenter of triangle LIV. Line G passes through point M parallel to segment IV. Point E is the reflection of point I across line G. Let Y be the midpoint of segment VD. Find the length of segment YE. 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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AI = 5. Let J be the rotation of point I by angle 7pi/12 around point A. Point R is the reflection of point J across segment IA. Point X is the orthocenter of triangle RAJ. Let E be the midpoint of segment AJ. What is the measure of angle EAX, 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 KH = 10. Point P is the reflection of point K across point H. Line O is the perpendicular bisector of segment PK. Let C be the rotation of point P by angle pi/12 around point K. Let G be the midpoint of segment CK. Let J be the midpoint of segment HC. Point U is the midpoint of segment CG. Let Y be the orthocenter of triangle PCU. Point T is the reflection of point Y across line O. Find the measure of angle YTJ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let XV = 2. Point Y is obtained by rotating point V by angle pi/2 about point X. Line A is the angle bisector of angle YVX. Point M is the reflection of point X across line A. Let G be the midpoint of segment VX. Point Z is the reflection of point M across segment XY. Find the distance between points G and Z. Express your answer as a decimal number rounded to 2 decimal places.

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Let UY = 10. Point N is obtained by rotating point U by angle pi/3 about point Y. Let A be the reflection of point Y across point U. Circle Q is the incircle of triangle ANY. Point S is the inversion of point A with respect to circle Q. What is the length of segment NS? Express your answer as a decimal number rounded to 2 decimal places.

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JV = 6. Point F is obtained by rotating point V by angle pi/6 about point J. Point S is the midpoint of segment FJ. Let N be the line through point J perpendicular to segment JF. Let Y be the line through point F parallel to line N. Point X is the reflection of point J across line Y. Let H be the rotation of point J by angle 2pi/3 around point S. Let B be the reflection of point X across point H. Point K is the reflection of point B across point V. Point P is the reflection of point K across segment JF. Point W is the midpoint of segment PJ. What is the measure of angle WVX, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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EZ = 11. Let D be the reflection of point E across point Z. Let B be the rotation of point D by angle 2pi/3 around point E. Let R be the perpendicular bisector of segment BZ. Point N is the reflection of point D across line R. Let H be the centroid of triangle EBN. What is the distance between points H and B? Express your answer as a decimal number rounded to 2 decimal places.

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NH = 1. Let M be the reflection of point H across point N. Point E is obtained by rotating point N by angle pi/6 about point M. Point S is obtained by rotating point E by angle 11pi/12 about point N. Point L is the reflection of point N across point E. Let F be the perpendicular bisector of segment MS. Line U is the perpendicular bisector of segment LN. Let Z be the reflection of line F across line U. Point Q is the reflection of point H across line U. Line T is the reflection of line F across point L. Lines Z and T intersect at point V. What is the measure of angle MQV, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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ED = 12. Let O be the midpoint of segment ED. Let B be the rotation of point O by angle pi/4 around point E. Let F be the reflection of point D across point B. Point I is the reflection of point F across point O. Let J be the centroid of triangle DEB. Find the circumradius of triangle BJI. Express your answer as a decimal number rounded to 2 decimal places.

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OK = 8. Line B is the perpendicular bisector of segment OK. Let T be the midpoint of segment OK. Line W is the reflection of line B across point O. Point P is on line W with |PK| = 4. Point I is the orthocenter of triangle TPO. Point Q is the reflection of point I across line B. Let N be the incircle of triangle TPO. Let R be the inversion of point K in circle N. Point L is the midpoint of segment OT. Point S is the reflection of point L across point R. Find the measure of angle SQR, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let LE = 1. Let N be the rotation of point E by angle pi/12 around point L. Point T is the reflection of point E across point N. Let W be the circumcenter of triangle NLE. Point Z is the reflection of point W across segment NL. Line I passes through points Z and W. Segment LE and line I intersect at point C. Find the distance between points T and C. Express your answer as a decimal number rounded to 2 decimal places.

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Let DK = 10. Let I be the reflection of point K across point D. Line B passes through point I perpendicular to segment KD. Let V be the rotation of point K by angle 7pi/12 around point D. Point M is the reflection of point D across line B. Let F be the line through point V parallel to segment KD. Point Z is the reflection of point K across line F. Let A be the reflection of point V across point M. What is the circumradius of triangle ZVA? Express your answer as a decimal number rounded to 2 decimal places.

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DF = 8. Point H is obtained by rotating point F by angle 3pi/4 about point D. Let M be the reflection of point H across point D. Point N is the midpoint of segment DH. Point X is the reflection of point H across segment FD. Point B is the reflection of point X across point M. Point I is the reflection of point B across segment NF. Find the measure of angle BHI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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ZW = 1. Let M be the rotation of point W by angle 2pi/3 around point Z. Point S is the orthocenter of triangle ZWM. Point F is obtained by rotating point S by angle pi/6 about point Z. Let J be the rotation of point F by angle 7pi/12 around point W. Line T is the perpendicular bisector of segment MZ. Point C is the midpoint of segment MS. Let V be the line through point C perpendicular to line T. Let X be the line through point F perpendicular to line V. Point H is the midpoint of segment WF. Point D is the reflection of point J across line X. Find the measure of angle HCD, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let GA = 6. Point P is the midpoint of segment GA. Point U is obtained by rotating point P by angle pi/4 about point G. Let J be the midpoint of segment UA. Let T be the midpoint of segment PJ. Point L is the reflection of point U across point P. What is the inradius of triangle LJT? Express your answer as a decimal number rounded to 2 decimal places.

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Let SE = 8. Point V is obtained by rotating point E by angle 5pi/6 about point S. Let G be the rotation of point V by angle 5pi/12 around point E. Let Y be the midpoint of segment SG. Point U is the reflection of point E across point S. Let W be the orthocenter of triangle UGY. What is the distance between points W and G? Express your answer as a decimal number rounded to 2 decimal places.

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Let VH = 8. Let Y be the rotation of point V by angle pi/4 around point H. Let C be the reflection of point Y across point V. Point B is obtained by rotating point Y by angle 11pi/12 about point C. Point K is the reflection of point V across point B. Let W be the midpoint of segment HV. Point N is the reflection of point B across point C. Find the circumradius of triangle NWK. Express your answer as a decimal number rounded to 2 decimal places.

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Let HJ = 6. Point I is obtained by rotating point H by angle 5pi/12 about point J. Let K be the orthocenter of triangle IHJ. Point Z is the reflection of point J across segment IH. Point X is the reflection of point Z across segment HJ. What is the distance between points K and X? Express your answer as a decimal number rounded to 2 decimal places.

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Let QU = 1. Point R is obtained by rotating point Q by angle 5pi/6 about point U. Point D is the reflection of point R across point Q. Point I is the midpoint of segment RQ. Point W is the midpoint of segment UD. What is the circumradius of triangle WIU? Express your answer as a decimal number rounded to 2 decimal places.

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MF = 9. Let J be the vector from point F to point M. Let A be point M translated by vector J. Let X be the rotation of point M by angle 5pi/6 around point A. Line T is the perpendicular bisector of segment AF. Point N is the midpoint of segment AX. Point G is the reflection of point X across line T. Let C be the reflection of point N across point G. Point Q is the circumcenter of triangle FCG. Find the distance between points C and Q. Express your answer as a decimal number rounded to 2 decimal places.

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Let UD = 12. Let K be the midpoint of segment DU. Let B be the rotation of point D by angle pi/3 around point K. Point J is the midpoint of segment DK. Let N be the rotation of point B by angle 7pi/12 around point D. Let L be the rotation of point D by angle 11pi/12 around point J. Let R be the perpendicular bisector of segment JL. Point P is the reflection of point N across line R. Let S be the line through point U parallel to segment ND. Point Z is the reflection of point D across line S. Find the circumradius of triangle PZL. Express your answer as a decimal number rounded to 2 decimal places.

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Let KY = 2. Let Q be the midpoint of segment KY. Point C is obtained by rotating point Q by angle 3pi/4 about point Y. Let L be the midpoint of segment YC. Let N be the orthocenter of triangle YCK. Let G be the circumcenter of triangle YCK. Let H be the line through points K and L. Point W is the incenter of triangle YCK. Let I be the reflection of line H through point W. Let Z be the reflection of point N across point G. Point J is the reflection of point Y across line I. What is the measure of angle JCZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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XP = 11. Let L be the vector from point X to point P. Let J be point P translated by vector L. Let U be the rotation of point J by angle 3pi/4 around point P. Let I be the reflection of point X across point U. Point A is the midpoint of segment IU. Let S be the reflection of point A across point J. What is the length of segment SI? Express your answer as a decimal number rounded to 2 decimal places.

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BU = 2. Point M is obtained by rotating point U by angle pi/6 about point B. Let P be the rotation of point B by angle 2pi/3 around point M. Let S be the perpendicular bisector of segment UB. Let O be the rotation of point U by angle pi/4 around point P. Line G passes through point O parallel to line S. Point R is the reflection of point P across line S. Line C passes through points M and R. Line Q is the reflection of line G across point R. Let H be the midpoint of segment BU. Let V be the reflection of line C across line Q. Point W is the reflection of point H across line V. Find the measure of angle PWO, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let DH = 7. Let X be the rotation of point H by angle 7pi/12 around point D. Point W is the midpoint of segment HD. Point C is the midpoint of segment WH. Point R is the circumcenter of triangle XCD. What is the measure of angle RXH, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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CD = 2. Point W is the reflection of point C across point D. Point N is obtained by rotating point W by angle pi/4 about point D. Let H be the centroid of triangle DCN. Line S is the perpendicular bisector of segment HC. Point Q is the reflection of point D across line S. Point U is the reflection of point Q across segment ND. What is the measure of angle WUQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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CZ = 2. Let O be the perpendicular bisector of segment ZC. Let W be the midpoint of segment CZ. Point M is obtained by rotating point Z by angle pi/4 about point C. Let X be the rotation of point C by angle pi/4 around point W. Point Y is obtained by rotating point X by angle pi/3 about point Z. Point F is the reflection of point X across point Y. Let V be the incircle of triangle CFX. Circle L is the reflection of circle V across line O. Let I be the inversion of point Z in circle L. Let H be the circumcenter of triangle CFX. Find the measure of angle MIH, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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ZN = 11. Let I be the rotation of point Z by angle pi/12 around point N. Point Y is the reflection of point N across segment ZI. Let V be the reflection of point Y across point Z. Line A passes through point V perpendicular to segment ZI. Line S passes through point V perpendicular to line A. Point L is the reflection of point Z across line S. Point U is the reflection of point Z across point L. Find the measure of angle UZN, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let AK = 3. Line I is the perpendicular bisector of segment KA. Let G be the rotation of point A by angle 11pi/12 around point K. Let V be the reflection of point G across point K. Point W is the reflection of point G across line I. Let P be the circumcircle of triangle AVW. Point T is the inversion of point G with respect to circle P. Let L be the midpoint of segment VW. What is the inradius of triangle LTK? Express your answer as a decimal number rounded to 2 decimal places.

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HD = 6. Let Z be the rotation of point H by angle 2pi/3 around point D. Let S be the orthocenter of triangle DZH. Let J be the reflection of point Z across point S. Let T be the midpoint of segment DJ. Find the length of segment ZT. Express your answer as a decimal number rounded to 2 decimal places.

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Let TW = 6. Point P is obtained by rotating point W by angle pi/4 about point T. Point I is the midpoint of segment PT. Point C is the midpoint of segment TW. Point A is the reflection of point P across segment TW. What is the circumradius of triangle ICA? Express your answer as a decimal number rounded to 2 decimal places.

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LI = 3. Point W is obtained by rotating point I by angle 5pi/6 about point L. Let X be the reflection of point W across point I. Point Y is the midpoint of segment XI. Point E is the reflection of point Y across point W. Find the circumradius of triangle ELW. Express your answer as a decimal number rounded to 2 decimal places.

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RQ = 5. Let J be the rotation of point R by angle pi/2 around point Q. Let D be the centroid of triangle RJQ. Point E is the reflection of point D across segment RJ. Let L be the midpoint of segment RQ. What is the distance between points L and E? Express your answer as a decimal number rounded to 2 decimal places.

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Let CP = 5. Point U is obtained by rotating point C by angle pi/4 about point P. Point V is the circumcenter of triangle PUC. Point H is the orthocenter of triangle PUC. Point D is the midpoint of segment HC. What is the distance between points V and D? Express your answer as a decimal number rounded to 2 decimal places.

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Let CW = 6. Let Y be the rotation of point W by angle pi/12 around point C. Let A be the circumcenter of triangle CWY. Let Z be the reflection of point C across point W. Let Q be the reflection of point A across point Z. What is the distance between points Y and Q? Express your answer as a decimal number rounded to 2 decimal places.

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Let AB = 7. Point Y is obtained by rotating point B by angle pi/2 about point A. Let L be the reflection of point A across point B. Let D be the midpoint of segment YB. Let C be the midpoint of segment LB. What is the length of segment YC? Express your answer as a decimal number rounded to 2 decimal places.

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CQ = 2. Let N be the midpoint of segment CQ. Let O be the vector from point Q to point N. Let Z be point C translated by vector O. Point K is obtained by rotating point Q by angle 11pi/12 about point Z. Point F is the reflection of point Q across segment NK. Point W is the reflection of point K across point Z. Circle M is the incircle of triangle FZK. Point Y is the inversion of point Z with respect to circle M. Point P is the reflection of point F across segment CQ. Find the measure of angle YWP, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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CB = 3. Point Q is the midpoint of segment CB. Point F is the midpoint of segment QB. Let P be the rotation of point Q by angle pi/6 around point F. Point D is the orthocenter of triangle CPQ. Point X is the reflection of point P across point D. Let S be the midpoint of segment XF. Let U be the midpoint of segment DS. What is the measure of angle QUD, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let CS = 4. Let O be the midpoint of segment SC. Point Z is obtained by rotating point O by angle 5pi/12 about point C. Line H is the angle bisector of angle SCZ. Let F be the reflection of point O across point S. Point X is the midpoint of segment FC. Point B is the reflection of point X across line H. What is the inradius of triangle XBZ? Express your answer as a decimal number rounded to 2 decimal places.

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CY = 12. Point X is the reflection of point C across point Y. Let O be the perpendicular bisector of segment YX. Point G is obtained by rotating point Y by angle pi/2 about point X. Point A is the reflection of point X across point G. Let U be the rotation of point G by angle 7pi/12 around point C. Point T is the circumcenter of triangle ACU. Let L be the midpoint of segment CU. Point S is the reflection of point T across line O. What is the distance between points L and S? Express your answer as a decimal number rounded to 2 decimal places.

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GC = 1. Let T be the reflection of point G across point C. Let D be the rotation of point T by angle pi/4 around point G. Line A passes through point T perpendicular to segment DC. Let E be the midpoint of segment DC. Point F is the reflection of point E across line A. Point U is the reflection of point D across point T. Let K be the midpoint of segment FC. Let P be the reflection of point K across point E. What is the measure of angle GPU, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let JN = 3. Let G be the perpendicular bisector of segment JN. Let S be the rotation of point N by angle 5pi/12 around point J. Line C is the reflection of line G across point J. Line E is the perpendicular bisector of segment SJ. Let K be the reflection of line E across line C. Point X is the reflection of point N across line K. Point T is the reflection of point X across segment JN. Point I is the reflection of point S across segment JN. What is the measure of angle IXT, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let HJ = 8. Point U is obtained by rotating point J by angle pi/12 about point H. Let S be the rotation of point H by angle pi/6 around point U. Point F is the midpoint of segment JU. Line X is the perpendicular bisector of segment SU. Point W is the reflection of point F across point S. Point G is the reflection of point J across line X. What is the length of segment WG? Express your answer as a decimal number rounded to 2 decimal places.

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VK = 2. Let G be the reflection of point V across point K. Point T is obtained by rotating point V by angle 7pi/12 about point G. Let O be the reflection of point T across point K. Point Q is the reflection of point T across point V. Point I is the reflection of point K across segment OQ. Point W is the reflection of point G across segment TO. What is the circumradius of triangle WQI? Express your answer as a decimal number rounded to 2 decimal places.

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Let TS = 11. Point B is obtained by rotating point T by angle pi/12 about point S. Point X is the midpoint of segment TB. Let Z be the centroid of triangle TBS. Point O is the reflection of point Z across segment TB. Let H be the reflection of point S across point O. Find the distance between points H and X. Express your answer as a decimal number rounded to 2 decimal places.

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AV = 6. Let K be the midpoint of segment AV. Point Q is obtained by rotating point V by angle 5pi/12 about point A. Let P be the angle bisector of angle AVQ. Let Z be the orthocenter of triangle QVK. Point B is the reflection of point Z across line P. Let F be the angle bisector of angle VQA. Let T be the circumcenter of triangle QVK. Point N is the reflection of point T across point B. Point L is the reflection of point N across line F. What is the distance between points L and N? Express your answer as a decimal number rounded to 2 decimal places.

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Let OH = 2. Point C is obtained by rotating point O by angle 11pi/12 about point H. Point Y is the midpoint of segment HC. Line X passes through points O and C. Point I is the reflection of point H across line X. What is the circumradius of triangle IYC? Express your answer as a decimal number rounded to 2 decimal places.

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Let SL = 3. Point R is obtained by rotating point L by angle pi about point S. Line T passes through point L perpendicular to segment LS. Point F is obtained by rotating point R by angle 3pi/4 about point L. Let I be the reflection of line T through point S. Let C be the perpendicular bisector of segment LS. Let D be the reflection of line C through point L. Let W be the rotation of point F by angle 7pi/12 around point S. Point P is the reflection of point S across point F. Line X is the reflection of line I across line D. Point H is the reflection of point W across line X. Find the circumradius of triangle PLH. Express your answer as a decimal number rounded to 2 decimal places.

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FE = 5. Point B is obtained by rotating point F by angle pi/4 about point E. Let I be the orthocenter of triangle BFE. Point V is the centroid of triangle BFE. Point N is the midpoint of segment BI. Find the distance between points V and N. Express your answer as a decimal number rounded to 2 decimal places.

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AX = 2. Let P be the rotation of point A by angle pi/2 around point X. Point N is obtained by rotating point A by angle pi/12 about point X. Point C is the orthocenter of triangle ANX. Let Z be the midpoint of segment PC. Let Y be the perpendicular bisector of segment AN. Point B is the reflection of point Z across line Y. Find the length of segment BZ. Express your answer as a decimal number rounded to 2 decimal places.

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Let KM = 11. Point C is obtained by rotating point K by angle 7pi/12 about point M. Point X is the circumcenter of triangle KCM. Let Z be the line through point X parallel to segment KC. Segment MK and line Z intersect at point L. Point H is the orthocenter of triangle KCM. Find the length of segment LH. Express your answer as a decimal number rounded to 2 decimal places.

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OR = 7. Let A be the midpoint of segment RO. Point I is obtained by rotating point R by angle 7pi/12 about point A. Let N be the angle bisector of angle IOA. Point D is the reflection of point I across line N. Let J be the line through point R perpendicular to line N. Point F is the reflection of point D across line J. Point T is the midpoint of segment AR. Find the inradius of triangle TAF. Express your answer as a decimal number rounded to 2 decimal places.

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DY = 9. Let N be the rotation of point Y by angle 5pi/6 around point D. Line O passes through point N parallel to segment YD. Point M is the midpoint of segment DY. Let A be the angle bisector of angle NDM. Let U be the perpendicular bisector of segment YD. Let I be the reflection of line O across line A. Line K is the reflection of line U across point D. Lines I and K intersect at point R. Find the inradius of triangle NMR. Express your answer as a decimal number rounded to 2 decimal places.

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MK = 8. Point J is the midpoint of segment KM. Point R is obtained by rotating point K by angle pi/4 about point M. Let C be the rotation of point R by angle pi/6 around point J. Let Y be the rotation of point K by angle pi/3 around point R. Line T is the angle bisector of angle YRC. Let L be the midpoint of segment CY. Point I is the reflection of point L across line T. Find the inradius of triangle KIL. Express your answer as a decimal number rounded to 2 decimal places.

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Let PJ = 8. Point C is obtained by rotating point P by angle pi/2 about point J. Circle Y is the circumcircle of triangle CPJ. Let I be the reflection of point P across point J. Let B be the reflection of point C across point J. Let S be the reflection of point J across point I. Let A be the midpoint of segment SC. Let X be the inversion of point A in circle Y. Let E be the angle bisector of angle PCX. Let M be the line through point S parallel to line E. Let T be the line through point B perpendicular to line M. Point W is the reflection of point A across line T. What is the distance between points W and I? Express your answer as a decimal number rounded to 2 decimal places.

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Let UX = 10. Let Y be the midpoint of segment UX. Point Q is the midpoint of segment YU. Let T be the perpendicular bisector of segment XU. Point L is obtained by rotating point Q by angle 11pi/12 about point U. Let V be the reflection of line T through point X. Point R is the reflection of point Q across line V. Let H be the midpoint of segment RL. Find the inradius of triangle HLQ. Express your answer as a decimal number rounded to 2 decimal places.

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Let UK = 3. Let G be the rotation of point K by angle 5pi/6 around point U. Point E is the midpoint of segment GK. Let N be the angle bisector of angle KEG. Point F is the midpoint of segment GU. Point Z is the reflection of point E across point F. Point J is the reflection of point Z across line N. What is the measure of angle FJK, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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TZ = 2. Let Y be the rotation of point T by angle pi/3 around point Z. Point G is the reflection of point Y across point T. Let S be the circumcenter of triangle YZG. Let M be the incircle of triangle YZG. Let V be the inversion of point S in circle M. What is the length of segment VS? Express your answer as a decimal number rounded to 2 decimal places.

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Let JD = 7. Let Z be the midpoint of segment DJ. Let I be the rotation of point J by angle pi/6 around point Z. Let O be the midpoint of segment JI. Let R be the circumcenter of triangle ODI. Let N be the midpoint of segment RD. Find the distance between points O and N. Express your answer as a decimal number rounded to 2 decimal places.

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Let EG = 2. Let R be the rotation of point E by angle 5pi/12 around point G. Let M be the rotation of point E by angle 3pi/4 around point G. Let A be the reflection of point R across point G. Let S be the reflection of point M across point G. Point P is the centroid of triangle SRE. What is the measure of angle GPA, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let PJ = 5. Point K is the reflection of point P across point J. Let Q be the rotation of point J by angle pi/2 around point P. Point X is the reflection of point K across segment JQ. Let E be the midpoint of segment JQ. Point F is the reflection of point Q across point X. Find the measure of angle EKF, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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PB = 8. Let Z be the rotation of point B by angle 5pi/6 around point P. Let G be the incircle of triangle BPZ. Let Y be the orthocenter of triangle BPZ. Point K is the inversion of point Z with respect to circle G. Let D be the midpoint of segment KY. Find the length of segment PD. Express your answer as a decimal number rounded to 2 decimal places.

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OB = 9. Let C be the rotation of point B by angle 7pi/12 around point O. Let M be the rotation of point C by angle pi/3 around point O. Let A be the rotation of point C by angle pi/12 around point B. Let H be the line through point B perpendicular to segment OB. Let S be the line through point C perpendicular to line H. Point I is the reflection of point M across line S. 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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Let TU = 5. Let N be the rotation of point U by angle 11pi/12 around point T. Let B be the perpendicular bisector of segment UT. Point M is the reflection of point N across line B. Let X be the reflection of line B through point N. Point H is the reflection of point T across line X. Point Y is the reflection of point M across segment HU. Find the distance between points Y and H. Express your answer as a decimal number rounded to 2 decimal places.

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NB = 9. Point K is the midpoint of segment NB. Let J be the midpoint of segment KN. Point M is obtained by rotating point J by angle 5pi/12 about point N. Point V is the reflection of point N across point M. Point L is the circumcenter of triangle VJK. Point R is the reflection of point B across point L. What is the distance between points R and K? Express your answer as a decimal number rounded to 2 decimal places.

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AU = 5. Point F is the midpoint of segment UA. Point N is obtained by rotating point F by angle pi/2 about point A. Point B is the centroid of triangle AFN. Line O is the perpendicular bisector of segment UA. Point K is the reflection of point B across line O. Point J is the orthocenter of triangle NKU. What is the length of segment AJ? Express your answer as a decimal number rounded to 2 decimal places.

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Let DR = 2. Point Z is obtained by rotating point D by angle pi/2 about point R. Let C be the circumcircle of triangle ZDR. Circle P is the reflection of circle C across point Z. Point U is the incenter of triangle ZDR. Let O be the inversion of point U in circle P. What is the measure of angle UOD, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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GL = 2. Line N is the perpendicular bisector of segment GL. Point R is on line N with |RL| = 5. Let U be the perpendicular bisector of segment RL. Let X be the incenter of triangle LRG. Let Y be the line through point L parallel to line U. Line A passes through point G parallel to line Y. Point H is the midpoint of segment RX. Line A and segment LR intersect at point M. Find the length of segment MH. Express your answer as a decimal number rounded to 2 decimal places.

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Let YK = 10. Let R be the rotation of point K by angle 3pi/4 around point Y. Point S is the midpoint of segment KY. Let Z be the incenter of triangle SKR. Let Q be the centroid of triangle SKR. Find the distance between points Q and Z. Express your answer as a decimal number rounded to 2 decimal places.

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TS = 9. Line K passes through points T and S. Point Q is obtained by rotating point S by angle pi/6 about point T. Let F be the rotation of point S by angle 5pi/12 around point T. Point C is the midpoint of segment TF. Line V passes through point C parallel to line K. Point P is obtained by rotating point Q by angle pi/6 about point T. Point E is the reflection of point P across line V. Point I is the reflection of point Q across segment SP. Point W is the reflection of point I across segment PT. Find the measure of angle WEQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let AH = 9. Let X be the midpoint of segment HA. Point G is obtained by rotating point X by angle 7pi/12 about point H. Point C is obtained by rotating point X by angle pi/2 about point H. Let W be the reflection of point C across point G. Point E is the midpoint of segment WA. Point R is the centroid of triangle GWA. Find the distance between points E and R. Express your answer as a decimal number rounded to 2 decimal places.

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RZ = 4. Point D is the midpoint of segment ZR. Let C be the reflection of point D across point Z. Point T is obtained by rotating point C by angle pi/6 about point D. Let Y be the orthocenter of triangle ZCT. Point V is the circumcenter of triangle ZCT. What is the inradius of triangle YVT? Express your answer as a decimal number rounded to 2 decimal places.

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TR = 6. Point S is obtained by rotating point R by angle 2pi/3 about point T. Point D is the reflection of point S across point R. Point Z is the midpoint of segment DT. Let U be the centroid of triangle TRS. What is the measure of angle ZTU, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Point U is the centroid of triangle M. Let W be the line through point U parallel to segment MB. Let E be the reflection of line W through point X. Point R is the reflection of point M across line E. What is the length of segment RB? Express your answer as a decimal number rounded to 2 decimal places.

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DN = 9. Point U is obtained by rotating point D by angle pi/3 about point N. Let S be the perpendicular bisector of segment UN. Point P is the reflection of point U across point D. Line E is the reflection of line S across point U. Line B is the reflection of line E across point D. Point C is the reflection of point D across line B. Let Z be the circumcenter of triangle NUP. Find the measure of angle DZC, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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ZM = 4. Let X be the rotation of point M by angle pi/12 around point Z. Let C be the centroid of triangle MZX. Point P is the reflection of point C across segment MZ. Let D be the circumcenter of triangle PZC. Find the length of segment DC. Express your answer as a decimal number rounded to 2 decimal places.

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SK = 12. Point H is obtained by rotating point K by angle 11pi/12 about point S. Point V is the midpoint of segment SK. Point B is the midpoint of segment KH. Point F is the reflection of point S across segment KH. Let Z be the midpoint of segment VS. What is the distance between points Z and F? Express your answer as a decimal number rounded to 2 decimal places.

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HK = 2. Point A is obtained by rotating point H by angle pi/6 about point K. Let V be the circumcircle of triangle AKH. Let R be the incenter of triangle AKH. Let D be the inversion of point R in circle V. Point L is the midpoint of segment AK. Find the distance between points L and D. Express your answer as a decimal number rounded to 2 decimal places.

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BI = 10. Point S is obtained by rotating point I by angle 3pi/4 about point B. Let Z be the midpoint of segment SI. Point U is the reflection of point Z across segment BS. Point K is the reflection of point I across segment BS. Find the measure of angle KUZ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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UL = 9. Let Z be the rotation of point U by angle pi/2 around point L. Point H is the midpoint of segment UZ. Line K passes through point Z perpendicular to segment LZ. Let B be the rotation of point L by angle 5pi/6 around point U. Line T passes through point U parallel to segment LZ. Point O is the reflection of point B across point H. Let P be the reflection of line T through point B. Point G is the reflection of point O across line P. Point X lies on line K. Let Y be the reflection of point H across point Z. Find the measure of angle YGX, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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SU = 6. Point F is obtained by rotating point S by angle pi/3 about point U. Let C be the rotation of point F by angle pi/12 around point U. Let G be the angle bisector of angle CFS. Point E is the reflection of point U across line G. Point D is the reflection of point E across segment US. Find the circumradius of triangle DEF. Express your answer as a decimal number rounded to 2 decimal places.

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Let IX = 7. Point B is obtained by rotating point I by angle 7pi/12 about point X. Let N be the midpoint of segment XI. Point W is the reflection of point X across segment BI. Let U be the perpendicular bisector of segment BW. Let Z be the angle bisector of angle BNX. Line H is the reflection of line Z across line U. Let K be the circumcenter of triangle WNX. Segment WN and line H intersect at point D. What is the length of segment DK? Express your answer as a decimal number rounded to 2 decimal places.

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JT = 4. Let L be the midpoint of segment TJ. Line X is the angle bisector of angle JLT. Point Z is the midpoint of segment TL. Point R is on line X with |RL| = 11. Line F passes through points Z and R. Line G is the reflection of line F across point T. Point N is the reflection of point R across line G. Let M be the midpoint of segment LN. Let A be the reflection of point M across point N. What is the inradius of triangle TAN? Express your answer as a decimal number rounded to 2 decimal places.

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ZO = 12. Point T is the reflection of point Z across point O. Point L is obtained by rotating point T by angle 5pi/6 about point O. Let G be the incircle of triangle TLO. Point U is the orthocenter of triangle TLO. Point V is the inversion of point L with respect to circle G. Find the measure of angle UZV, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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GT = 6. Point V is obtained by rotating point T by angle 5pi/12 about point G. Let B be the rotation of point V by angle 5pi/6 around point G. Let H be the midpoint of segment TV. Point Z is the midpoint of segment VB. Let P be the centroid of triangle ZHG. What is the distance between points Z and P? Express your answer as a decimal number rounded to 2 decimal places.

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Let FS = 10. Let E be the vector from point S to point F. Let Q be the midpoint of segment FS. Let U be the rotation of point Q by angle pi/2 around point F. Point A is obtained by rotating point S by angle 11pi/12 about point F. Line X is the angle bisector of angle AQF. Let K be the reflection of line X through point U. Point O is the translation of point A by vector E. Line R passes through point O perpendicular to line X. Point V is the midpoint of segment QS. Let T be the reflection of line R through point Q. Point Z is the intersection of lines T and K. Point C is the orthocenter of triangle OZV. What is the length of segment FC? Express your answer as a decimal number rounded to 2 decimal places.

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Let LN = 10. Point O is the reflection of point N across point L. Let G be the perpendicular bisector of segment NO. Let V be the vector from point O to point N. Point D is the translation of point L by vector V. Point K is on line G with |KL| = 11. Let B be the line through points K and N. Point Q is the reflection of point L across line B. Let E be the midpoint of segment DK. Let C be the centroid of triangle QNE. Find the measure of angle CLE, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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BC = 7. Point G is the reflection of point B across point C. Point U is obtained by rotating point C by angle 11pi/12 about point B. Let F be the rotation of point B by angle pi/4 around point C. Let Y be the midpoint of segment FG. Let P be the orthocenter of triangle UBY. Let X be the incenter of triangle UBY. Find the length of segment PX. Express your answer as a decimal number rounded to 2 decimal places.

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Let EA = 3. Let K be the rotation of point A by angle 11pi/12 around point E. Let W be the midpoint of segment AK. Point Z is obtained by rotating point K by angle 11pi/12 about point W. Point T is obtained by rotating point E by angle 5pi/6 about point W. Let S be the rotation of point Z by angle pi/2 around point E. Point O is the reflection of point T across point A. Point J is the reflection of point S across segment EK. Point V is the midpoint of segment OE. What is the measure of angle JTV, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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CU = 11. Point D is obtained by rotating point U by angle 5pi/6 about point C. Let H be the centroid of triangle CDU. Point F is the reflection of point C across point D. Let P be the line through points H and F. Line P and segment UC intersect at point V. Point O is the reflection of point V across segment CD. What is the length of segment FO? Express your answer as a decimal number rounded to 2 decimal places.

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KE = 5. Let C be the rotation of point K by angle 5pi/6 around point E. Let M be the centroid of triangle CEK. Point G is the orthocenter of triangle CEK. Let N be the reflection of point M across point C. Let Q be the midpoint of segment MG. What is the measure of angle NKQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let AM = 9. Let Z be the rotation of point M by angle pi/3 around point A. Let F be the circumcenter of triangle MAZ. Point I is the reflection of point F across segment ZM. Point O is the reflection of point I across segment MA. What is the distance between points Z and O? Express your answer as a decimal number rounded to 2 decimal places.

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Let LR = 3. Let Z be the rotation of point R by angle pi/12 around point L. Let Y be the midpoint of segment LR. Point W is the orthocenter of triangle LZY. Let T be the midpoint of segment WR. What is the circumradius of triangle ZTW? Express your answer as a decimal number rounded to 2 decimal places.

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BP = 1. Point S is obtained by rotating point P by angle pi about point B. Let K be the rotation of point S by angle pi/12 around point P. Let G be the midpoint of segment SK. Let X be the circumcenter of triangle BKG. What is the circumradius of triangle KXP? Express your answer as a decimal number rounded to 2 decimal places.

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Let GX = 2. Let H be the line through point G perpendicular to segment GX. Let E be the perpendicular bisector of segment GX. Point W is on line E with |WX| = 5. Point C is the midpoint of segment WX. Point L is the reflection of point G across point C. Let Q be the reflection of line E across line H. Point A is the midpoint of segment GX. Point I is the reflection of point C across line Q. What is the measure of angle LIA, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let LI = 4. Let Q be the rotation of point L by angle 3pi/4 around point I. Line F is the perpendicular bisector of segment IL. Point H is the midpoint of segment IQ. Point D is the reflection of point H across line F. What is the circumradius of triangle DIH? Express your answer as a decimal number rounded to 2 decimal places.

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IX = 6. Point D is obtained by rotating point I by angle pi/6 about point X. Let T be the reflection of point I across point D. Point H is the midpoint of segment XT. Let A be the midpoint of segment HT. Find the inradius of triangle AXD. Express your answer as a decimal number rounded to 2 decimal places.

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Let WD = 8. Point G is the midpoint of segment WD. Point N is obtained by rotating point G by angle pi/2 about point D. Let A be the reflection of point G across point N. Let J be the midpoint of segment WN. Point U is the incenter of triangle AWG. What is the measure of angle UJN, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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DU = 4. Point W is obtained by rotating point D by angle 11pi/12 about point U. Let J be the midpoint of segment DW. Let C be the midpoint of segment UW. Let G be the midpoint of segment CJ. What is the inradius of triangle GUW? Express your answer as a decimal number rounded to 2 decimal places.

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GT = 1. Let R be the rotation of point T by angle 11pi/12 around point G. Point K is the circumcenter of triangle RGT. Point A is the incenter of triangle RGT. Point S is the midpoint of segment RT. Point J is the midpoint of segment AS. Line F is the perpendicular bisector of segment GT. Point Z is the reflection of point S across point K. Point D is the reflection of point Z across line F. Find the distance between points D and J. Express your answer as a decimal number rounded to 2 decimal places.

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CN = 1. Let F be the rotation of point N by angle 7pi/12 around point C. Point J is the orthocenter of triangle CNF. Let H be the circumcenter of triangle CNF. Let E be the reflection of point N across point C. Let V be the reflection of point J across point N. Find the measure of angle VHE, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let GD = 8. Let Z be the rotation of point D by angle 7pi/12 around point G. Point I is the midpoint of segment DZ. Let N be the line through point I parallel to segment GD. Let W be the centroid of triangle ZGD. Point V is the reflection of point W across line N. What is the distance between points V and I? Express your answer as a decimal number rounded to 2 decimal places.

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Line X is the perpendicular bisector of segment O17D. Let Z be the line through point T parallel to line X. Point F is the reflection of point T across segment O17D. Point C is the reflection of point T across line Z. Let M be the circumcenter of triangle D. Find the measure of angle MCF, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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HS = 2. Let Z be the perpendicular bisector of segment HS. Point K lies on line Z at distance 4 from point S. Point G is the midpoint of segment HK. Point N is the centroid of triangle HKS. Line C is the perpendicular bisector of segment NK. Point F is the reflection of point K across point H. Point D is the reflection of point G across line C. Point Y is the reflection of point F across segment KS. Find the distance between points D and Y. Express your answer as a decimal number rounded to 2 decimal places.

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OJ = 9. Let T be the rotation of point O by angle pi/4 around point J. Let H be the reflection of point O across point J. Line F is the angle bisector of angle HOJ. Point Y is the reflection of point T across line F. Find the circumradius of triangle OYT. Express your answer as a decimal number rounded to 2 decimal places.

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Let WZ = 10. Let K be the midpoint of segment ZW. Point O is obtained by rotating point K by angle pi/3 about point Z. Point L is the reflection of point W across point O. Let C be the reflection of point O across point K. Let G be the midpoint of segment LC. Let D be the reflection of point W across point L. Find the inradius of triangle WDG. Express your answer as a decimal number rounded to 2 decimal places.

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EC = 10. Point A is obtained by rotating point C by angle 7pi/12 about point E. Circle W is the incircle of triangle AEC. Let O be the centroid of triangle AEC. Let M be the inversion of point O in circle W. Find the length of segment MA. Express your answer as a decimal number rounded to 2 decimal places.

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Let LF = 12. LN = 12. Point S is obtained by rotating point L by angle 2pi/3 about point F. Point Q is the midpoint of segment FL. Let U be the circumcenter of triangle FLN. Point J is the reflection of point S across point U. Point G is the midpoint of segment QJ. What is the length of segment UG? Express your answer as a decimal number rounded to 2 decimal places.

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HU = 7. Point G is obtained by rotating point H by angle 7pi/12 about point U. Let P be the midpoint of segment HG. Point J is the reflection of point P across segment GU. Point E is the orthocenter of triangle UHG. What is the measure of angle EUJ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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AF = 9. Line S passes through points A and F. Point D is obtained by rotating point A by angle 11pi/12 about point F. Point O is the reflection of point D across line S. Let E be the line through point F perpendicular to line S. Point V is on line E. Point I is the reflection of point D across point A. Point X is the reflection of point F across point D. Point N is the midpoint of segment XO. Find the measure of angle AIN, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let TB = 9. Point V is obtained by rotating point T by angle pi/12 about point B. Point L is the centroid of triangle BTV. Point Z is the midpoint of segment LB. Point W is the reflection of point L across segment TV. What is the distance between points W and Z? Express your answer as a decimal number rounded to 2 decimal places.

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WX = 9. Point P is obtained by rotating point X by angle 5pi/6 about point W. Line L passes through point W perpendicular to segment WX. Let A be the reflection of point X across point P. Line T is the perpendicular bisector of segment WP. Let E be the reflection of line L across line T. Point F is the reflection of point A across line E. Point Q is the reflection of point F across segment AW. Find the measure of angle XQA, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let GV = 3. Point H is obtained by rotating point V by angle pi/2 about point G. Point Q is the midpoint of segment VH. Let E be the midpoint of segment VG. Point U is the midpoint of segment HE. What is the length of segment QU? Express your answer as a decimal number rounded to 2 decimal places.

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Let ZK = 11. Point M is the reflection of point Z across point K. Let V be the rotation of point M by angle 5pi/6 around point K. Point O is the reflection of point V across point K. Let I be the midpoint of segment VZ. Point F is the reflection of point I across segment VM. What is the distance between points F and V? Express your answer as a decimal number rounded to 2 decimal places.

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TK = 7. Point C is the reflection of point T across point K. Let G be the rotation of point T by angle pi/3 around point C. Let B be the angle bisector of angle GKT. Point W is the reflection of point G across line B. Let H be the line through points T and K. Line N passes through point G parallel to line H. Point E is the reflection of point T across point G. Let M be the reflection of line N through point E. Point D is the reflection of point W across line M. Find the measure of angle TGD, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let OH = 8. Point Z is the reflection of point H across point O. Let I be the midpoint of segment HO. Let M be the reflection of point O across point Z. Let U be the rotation of point I by angle 7pi/12 around point H. Point R is obtained by rotating point U by angle pi/12 about point H. Let W be the midpoint of segment ZM. Let L be the line through point Z parallel to segment RW. Point A is the intersection of segment OR and line L. Find the distance between points A and R. Express your answer as a decimal number rounded to 2 decimal places.

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VM = 2. Line R passes through points V and M. Let K be the reflection of point M across point V. Point I is obtained by rotating point V by angle pi/6 about point K. Let Z be the midpoint of segment MV. Let G be the reflection of point V across point I. Let O be the reflection of point I across point Z. Let P be the midpoint of segment IK. Point J is the reflection of point O across line R. Point L is the reflection of point J across segment GZ. What is the circumradius of triangle OLJ? Express your answer as a decimal number rounded to 2 decimal places.

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Let FT = 1. Point G is obtained by rotating point F by angle 3pi/4 about point T. Let L be the midpoint of segment TF. Let U be the incenter of triangle FTG. What is the circumradius of triangle ULG? Express your answer as a decimal number rounded to 2 decimal places.

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Let IX = 5. Let N be the rotation of point I by angle pi/3 around point X. Let G be the circumcenter of triangle XIN. Let E be the midpoint of segment XI. Let V be the line through point E perpendicular to segment IN. Point B is the reflection of point I across line V. Point P is the reflection of point G across line V. Find the length of segment BP. Express your answer as a decimal number rounded to 2 decimal places.

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QP = 4. Let O be the rotation of point P by angle 5pi/6 around point Q. Point V is the centroid of triangle OPQ. Point S is the reflection of point V across point O. Let Z be the reflection of point S across point V. Find the distance between points Z and Q. Express your answer as a decimal number rounded to 2 decimal places.

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Let WL = 7. Point B is the midpoint of segment WL. Let V be the midpoint of segment WB. Let Q be the angle bisector of angle WVL. Point M is the reflection of point L across point W. Let O be the rotation of point B by angle 7pi/12 around point M. Point P is the reflection of point M across line Q. Point H is the reflection of point P across segment OV. Point F lies on segment OV. Find the measure of angle MHF, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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YV = 9. Let K be the vector from point Y to point V. Let X be point V translated by vector K. Let P be the midpoint of segment VX. Let E be the rotation of point P by angle pi/2 around point Y. Let D be the reflection of point V across point E. Let M be the incenter of triangle DPE. Let S be the circumcenter of triangle DPE. Let J be the midpoint of segment ED. What is the measure of angle MJS, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let GF = 11. Let J be the rotation of point F by angle pi/4 around point G. Point B is the reflection of point G across point J. Point C is the orthocenter of triangle GBF. Point H is the midpoint of segment FG. What is the measure of angle CHB, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let ZG = 1. Let R be the rotation of point Z by angle pi/6 around point G. Let N be the midpoint of segment ZR. Point P is the midpoint of segment NZ. Point J is the reflection of point G across point Z. Let U be the circumcenter of triangle RPG. What is the length of segment JU? Express your answer as a decimal number rounded to 2 decimal places.

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Let QL = 5. Let K be the rotation of point Q by angle pi/3 around point L. Let Y be the rotation of point K by angle 7pi/12 around point L. Point W is the reflection of point L across segment KY. Let A be the midpoint of segment KW. Let C be the circumcircle of triangle YQL. Point X is the inversion of point K with respect to circle C. What is the length of segment XA? Express your answer as a decimal number rounded to 2 decimal places.

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IR = 2. Let Y be the line through points R and I. Let V be the line through point R perpendicular to line Y. Point Z is on line V. Point X is the midpoint of segment IZ. Let H be the midpoint of segment IR. Point N is the centroid of triangle HXZ. What is the distance between points N and X? Express your answer as a decimal number rounded to 2 decimal places.

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Let RN = 9. Let H be the rotation of point R by angle pi/2 around point N. Point U is the midpoint of segment HN. Let Y be the orthocenter of triangle RUH. Let M be the circumcircle of triangle RUH. Line B passes through point R perpendicular to segment HR. Point S is the reflection of point N across line B. Let L be the inversion of point N in circle M. What is the measure of angle SYL, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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DV = 6. Let M be the reflection of point V across point D. Let A be the perpendicular bisector of segment DM. Let H be the vector from point M to point V. Point P is on line A with |PD| = 10. Let W be point P translated by vector H. Point U is the midpoint of segment VP. Let O be the line through point P perpendicular to segment VP. Point F is the reflection of point U across line O. Let S be the reflection of point D across point W. Find the measure of angle SDF, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let MP = 9. Let K be the rotation of point P by angle 3pi/4 around point M. Point N is obtained by rotating point P by angle 2pi/3 about point K. Point Q is obtained by rotating point N by angle 5pi/12 about point P. Let O be the rotation of point M by angle pi/6 around point N. Point Y is the midpoint of segment OQ. Let X be the circumcenter of triangle OQP. Find the circumradius of triangle YNX. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let NQ = 3. Point H is obtained by rotating point N by angle pi/12 about point Q. Point E is the reflection of point Q across segment NH. Point F is the incenter of triangle NHQ. Let P be the reflection of point E across point H. Point W is the reflection of point P across point F. Let Z be the orthocenter of triangle NEW. Find the measure of angle FWZ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let IY = 12. Let O be the rotation of point Y by angle pi/3 around point I. Point B is the midpoint of segment OI. Point J is the centroid of triangle BOY. Point M is the midpoint of segment OY. Find the measure of angle IMJ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
LO = 1. Point F is the reflection of point O across point L. Point P is obtained by rotating point F by angle 2pi/3 about point L. Let W be the rotation of point L by angle pi/2 around point F. Point T is the midpoint of segment FL. Point U is the reflection of point W across point O. Point M is the reflection of point W across point T. Point J is the reflection of point U across point M. Point Q is the incenter of triangle PJF. Find the measure of angle QOU, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
XQ = 4. Point H is the reflection of point X across point Q. Let Y be the rotation of point H by angle pi/12 around point X. Let M be the midpoint of segment XQ. Point C is the reflection of point H across segment YX. What is the inradius of triangle MYC? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let IV = 1. Let C be the rotation of point I by angle 5pi/6 around point V. Let H be the orthocenter of triangle IVC. Point B is the reflection of point C across point V. Let R be the midpoint of segment CI. Point Z is the reflection of point B across point R. What is the distance between points Z and H? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
QF = 7. Let Y be the vector from point Q to point F. Point E is the translation of point F by vector Y. Let M be the rotation of point F by angle 5pi/12 around point Q. Line L is the perpendicular bisector of segment MF. Point U is the reflection of point E across line L. Find the circumradius of triangle UEM. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
NB = 9. Let J be the rotation of point B by angle pi/12 around point N. Point H is the incenter of triangle NBJ. Point E is on segment BJ. Point I is the midpoint of segment HE. Point W is the midpoint of segment EN. What is the distance between points I and W? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
FQ = 8. Point E is obtained by rotating point F by angle 5pi/6 about point Q. Let H be the reflection of point Q across point F. Point N is the reflection of point H across segment EQ. Let R be the midpoint of segment EN. Let B be the midpoint of segment EQ. Line O passes through point N perpendicular to segment EQ. Point J is the reflection of point E across line O. Let A be the incircle of triangle BRN. Let P be the inversion of point R in circle A. Find the distance between points P and J. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
QH = 1. Let N be the rotation of point Q by angle pi/2 around point H. Point V is the midpoint of segment HN. Let G be the circumcenter of triangle HVQ. Point F is the incenter of triangle HVQ. Find the measure of angle QFG, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ZP = 9. Let B be the rotation of point Z by angle 5pi/6 around point P. Point W is the orthocenter of triangle BPZ. Point S is the centroid of triangle BPZ. Point Y is the incenter of triangle BPZ. Point U is the reflection of point S across segment ZB. Point J is the midpoint of segment YU. Find the measure of angle JWZ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let WM = 7. Point K is obtained by rotating point W by angle 2pi/3 about point M. Point A is the reflection of point M across point K. Point L is the reflection of point A across segment MW. Let N be the incenter of triangle ALW. Find the distance between points L and N. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let EX = 1. Point G is obtained by rotating point E by angle 3pi/4 about point X. Let K be the reflection of point E across point G. Point M is the orthocenter of triangle XKG. Point B is the reflection of point E across point M. What is the distance between points B and K? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let ME = 11. Let Y be the rotation of point M by angle 11pi/12 around point E. Point T is the circumcenter of triangle MYE. Let I be the orthocenter of triangle MYE. Point J is the reflection of point T across segment EM. Point O is the midpoint of segment YE. Find the measure of angle OJI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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