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OV = 3. Point G is obtained by rotating point O by angle pi/2 about point V. Let F be the reflection of point G across point V. Let T be the circumcenter of triangle GOF. Let W be the line through points F and T. Let C be the circumcircle of triangle GOF. Let N be the rotation of point T by angle 11pi/12 around point O. Circle Y is the reflection of circle C across line W. Point M is the reflection of point T across segment FG. Let H be the reflection of circle Y through point M. Let D be the reflection of point G across point T. Point K is the inversion of point N with respect to circle H. What is the distance between points D and K? Express your answer as a decimal number rounded to 2 decimal places.

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ZW = 9. Let E be the rotation of point W by angle 3pi/4 around point Z. Let T be the perpendicular bisector of segment WE. Point G is the midpoint of segment EZ. Point Q is the reflection of point G across line T. Point N is the orthocenter of triangle WEZ. Find the distance between points Q and N. Express your answer as a decimal number rounded to 2 decimal places.

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Let JU = 7. Let C be the rotation of point U by angle pi/2 around point J. Let T be the rotation of point J by angle pi/12 around point U. Let L be the midpoint of segment UT. Point V is the reflection of point L across segment TC. Let O be the centroid of triangle TCU. What is the distance between points O and V? Express your answer as a decimal number rounded to 2 decimal places.

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GO = 5. Line J is the perpendicular bisector of segment OG. Let Z be the line through point O parallel to line J. Let P be the reflection of line J across line Z. Point C is obtained by rotating point O by angle pi/3 about point G. Let Q be the rotation of point O by angle 5pi/12 around point G. Let D be the reflection of line Z across line P. Point X is the orthocenter of triangle GCO. Let R be the line through points Q and X. Point U is the intersection of lines D and R. Find the length of segment CU. Express your answer as a decimal number rounded to 2 decimal places.

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Let PV = 4. Let Z be the rotation of point P by angle pi/12 around point V. Line J is the perpendicular bisector of segment ZV. Let B be the line through point Z parallel to line J. Line G passes through point V parallel to line B. Point C is the reflection of point P across line G. Let R be the orthocenter of triangle CZP. Point D is the midpoint of segment CZ. What is the measure of angle RDP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let CX = 3. Point J is obtained by rotating point C by angle pi/2 about point X. Point H is the midpoint of segment XC. Point A is obtained by rotating point H by angle 5pi/6 about point C. Let F be the midpoint of segment XJ. Let G be the reflection of point F across point A. Let Z be the circumcenter of triangle GCA. Find the circumradius of triangle XGZ. Express your answer as a decimal number rounded to 2 decimal places.

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Let YA = 4. Point E is obtained by rotating point A by angle pi/6 about point Y. Let V be the rotation of point Y by angle 5pi/12 around point E. Let G be the rotation of point V by angle pi/12 around point Y. Let I be the midpoint of segment YG. Let N be the circumcenter of triangle YIV. Point R is the reflection of point N across segment VY. What is the distance between points Y and R? Express your answer as a decimal number rounded to 2 decimal places.

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Let TC = 5. Point V is obtained by rotating point T by angle pi/3 about point C. Point K is the midpoint of segment CV. Let H be the circumcenter of triangle KTV. Let B be the midpoint of segment HV. What is the distance between points B and V? Express your answer as a decimal number rounded to 2 decimal places.

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FJ = 8. Let K be the reflection of point F across point J. Point O is obtained by rotating point F by angle pi/4 about point K. Let W be the midpoint of segment JK. Let Y be the midpoint of segment WK. Point E is obtained by rotating point F by angle pi/3 about point K. Point A is the reflection of point E across point W. Let P be the reflection of point E across point Y. Let T be the incenter of triangle EOK. Point Z is the midpoint of segment TO. Find the measure of angle ZPA, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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UM = 2. Let H be the perpendicular bisector of segment MU. Point N is obtained by rotating point U by angle 5pi/12 about point M. Point O is the reflection of point N across line H. Point W is the incenter of triangle MNO. Let J be the circumcenter of triangle MNO. Find the measure of angle WOJ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let OH = 8. Let U be the perpendicular bisector of segment OH. Point G is on line U with |GH| = 3. Let W be the reflection of point O across point G. Let C be the orthocenter of triangle HGO. Let N be the reflection of point H across point C. Find the measure of angle NCW, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Point E is the reflection of point B across segment YY40. Point V is obtained by rotating point B by angle pi about point E. Let Q be the angle bisector of angle DBE. Circle J is the circumcircle of triangle D. Circle X is the reflection of circle J across line Q. Point C is the inversion of point V with respect to circle X. What is the length of segment CV? Express your answer as a decimal number rounded to 2 decimal places.

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SC = 12. Point A is obtained by rotating point C by angle 2pi/3 about point S. Point P is the centroid of triangle ASC. Line B passes through point S perpendicular to segment SC. Point G is the reflection of point P across line B. Find the distance between points P and G. Express your answer as a decimal number rounded to 2 decimal places.

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Let BH = 9. Line J is the perpendicular bisector of segment HB. Let O be the line through point B perpendicular to line J. Point A is the midpoint of segment HB. Let X be the midpoint of segment HA. Let I be the rotation of point A by angle pi/6 around point X. Point U is obtained by rotating point X by angle pi/3 about point H. Point E is obtained by rotating point A by angle 5pi/6 about point U. Point S is the reflection of point E across line O. Line K is the angle bisector of angle HSB. Point T is the reflection of point U across line K. Point V is the centroid of triangle EIH. What is the length of segment TV? Express your answer as a decimal number rounded to 2 decimal places.

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DB = 6. Let S be the rotation of point B by angle 3pi/4 around point D. Point W is the reflection of point S across point D. Let H be the reflection of point W across point S. Let C be the centroid of triangle SBD. What is the distance between points H and C? Express your answer as a decimal number rounded to 2 decimal places.

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JH = 12. JA = 1. Let D be the rotation of point A by angle 3pi/4 around point H. Point O lies on segment HA. Let L be the line through point O parallel to segment DH. Let W be the incenter of triangle ADH. Let Z be the line through point W parallel to line L. Segment HA and line Z intersect at point G. Find the measure of angle WGA, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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LQ = 8. Let H be the vector from point Q to point L. Point E is the translation of point L by vector H. Let I be the line through points E and Q. Line O passes through point E perpendicular to line I. Point V is on line O with |VQ| = 10. Let U be the perpendicular bisector of segment QV. Point Y is the reflection of point L across line U. Point X is the reflection of point L across point Q. Point J is the reflection of point Y across line O. Point N is the reflection of point J across point V. What is the measure of angle XNJ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let QL = 11. Point M is obtained by rotating point L by angle pi/2 about point Q. Let C be the rotation of point Q by angle pi/6 around point M. Point K is obtained by rotating point M by angle 5pi/12 about point L. Let R be the incircle of triangle QKM. Point G is the midpoint of segment MQ. Let W be the reflection of point M across point C. Let O be the inversion of point G in circle R. Point Y is the reflection of point C across point W. What is the measure of angle QYO, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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GB = 7. Point I is obtained by rotating point G by angle 2pi/3 about point B. Let U be the rotation of point B by angle pi/2 around point G. Let C be the midpoint of segment GU. Point L is the incenter of triangle GUI. Line K passes through points L and G. Point E is the intersection of segment UI and line K. What is the length of segment CE? Express your answer as a decimal number rounded to 2 decimal places.

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Let HN = 6. Point S is the midpoint of segment NH. Point P is the midpoint of segment SN. Point A is the reflection of point S across point N. Point L is the midpoint of segment AN. Point W is obtained by rotating point H by angle 2pi/3 about point S. Let V be the reflection of point S across point L. Let J be the reflection of point P across point W. Point T is the reflection of point V across segment HJ. Point K is the midpoint of segment HW. Find the measure of angle HKT, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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OE = 9. Let A be the perpendicular bisector of segment OE. Let Q be the reflection of line A through point O. Let R be the rotation of point E by angle 11pi/12 around point O. Let T be the reflection of line Q across line A. Let Y be the rotation of point R by angle pi/6 around point E. Point P is the reflection of point O across line T. Line M is the angle bisector of angle ROE. Line C is the reflection of line M across point R. Point V is the reflection of point Y across segment EO. Point N is the intersection of lines C and T. Point J is the reflection of point N across segment EO. Point K is the reflection of point V across point P. What is the measure of angle KJE, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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FE = 1. Point A is obtained by rotating point E by angle pi/2 about point F. Point I is obtained by rotating point A by angle pi/6 about point F. Line V is the angle bisector of angle FAI. Point M is the midpoint of segment EF. Let L be the perpendicular bisector of segment FI. Let G be the circumcircle of triangle EFA. Point X is the reflection of point E across line L. Let T be the reflection of circle G across line V. Let K be the inversion of point E in circle T. Let N be the reflection of point M across point K. Point R is the midpoint of segment XE. Find the length of segment RN. Express your answer as a decimal number rounded to 2 decimal places.

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Let FR = 6. Let Y be the rotation of point F by angle 2pi/3 around point R. Let W be the reflection of point F across point R. Point C is the reflection of point F across segment RY. Let V be the centroid of triangle RYF. What is the circumradius of triangle VCW? Express your answer as a decimal number rounded to 2 decimal places.

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Let VU = 9. Point J is the midpoint of segment VU. Point O is the reflection of point V across point U. Let D be the rotation of point V by angle pi/6 around point J. Let A be the perpendicular bisector of segment OJ. Point X is the reflection of point V across segment OD. Point T is the reflection of point U across segment OD. Point H is the reflection of point U across line A. What is the inradius of triangle XTH? Express your answer as a decimal number rounded to 2 decimal places.

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Let LW = 9. Point D is obtained by rotating point W by angle 5pi/12 about point L. Let R be the angle bisector of angle DLW. Let K be the line through point W parallel to line R. Point Z is the reflection of point L across line K. Let G be the orthocenter of triangle LDW. What is the measure of angle LZG, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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YK = 10. Line C is the perpendicular bisector of segment YK. Point G lies on line C at distance 12 from point K. Point J is the midpoint of segment KG. Let T be the midpoint of segment JY. Circle B is the circumcircle of triangle GJY. Point A is the midpoint of segment YT. Let F be the inversion of point A in circle B. Let P be the centroid of triangle GJY. Find the distance between points F and P. Express your answer as a decimal number rounded to 2 decimal places.

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Let AI = 6. Point J is the reflection of point I across point A. Let B be the line through points A and J. Point D is obtained by rotating point I by angle pi/3 about point J. Point T is the reflection of point D across point I. Line R is the reflection of line B across point T. Line K is the reflection of line R across line B. Point H is the reflection of point I across line K. Point O is the reflection of point H across line R. What is the measure of angle DHO, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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LH = 6. Line R is the perpendicular bisector of segment LH. Let Y be the vector from point H to point L. Point X is the reflection of point L across point H. Point G is on line R with |GX| = 8. Line S passes through points X and L. Let M be point G translated by vector Y. Point D lies on line S at distance 6 from point M. Let J be the midpoint of segment MX. Let C be the midpoint of segment GL. Let W be the reflection of point D across point X. Find the measure of angle CWJ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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XJ = 5. Point K is obtained by rotating point J by angle 3pi/4 about point X. Line A is the angle bisector of angle XKJ. Point N is on line A. Point R is the reflection of point K across segment XN. Find the length of segment RX. Express your answer as a decimal number rounded to 2 decimal places.

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Let UO = 1. Point B is obtained by rotating point U by angle pi/6 about point O. Let K be the reflection of point B across point U. Let F be the orthocenter of triangle OBK. Let G be the midpoint of segment OF. Let P be the incenter of triangle OBK. Find the distance between points G and P. Express your answer as a decimal number rounded to 2 decimal places.

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FY = 5. Point S is obtained by rotating point F by angle pi/6 about point Y. Point A is obtained by rotating point Y by angle 3pi/4 about point S. Let M be the midpoint of segment FA. Point Q is the midpoint of segment AS. Point N is the midpoint of segment MQ. Find the length of segment NS. Express your answer as a decimal number rounded to 2 decimal places.

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WT = 3. Point X is obtained by rotating point T by angle pi/4 about point W. Let K be the incenter of triangle WTX. Line V passes through points K and W. Point M is the intersection of segment TX and line V. Find the distance between points M and W. Express your answer as a decimal number rounded to 2 decimal places.

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Let AS = 2. Let W be the midpoint of segment SA. Let H be the line through points W and A. Let R be the line through point A perpendicular to line H. Line G passes through point W parallel to line R. Point V is on line G with |VS| = 4. Point T is the midpoint of segment VS. Let C be the reflection of line R across line G. Let U be the midpoint of segment SW. Point O is the reflection of point U across line R. Point Q is the reflection of point T across line C. Find the measure of angle OVQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let LP = 11. Let M be the rotation of point L by angle 7pi/12 around point P. Point B is the midpoint of segment ML. Let F be the rotation of point B by angle pi/2 around point L. Point Q is the incenter of triangle LMF. Point T is the midpoint of segment FL. Let I be the centroid of triangle LMF. Find the measure of angle TQI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let FZ = 11. Let R be the midpoint of segment FZ. Point W is obtained by rotating point R by angle 5pi/6 about point Z. Let H be the midpoint of segment RW. Let K be the rotation of point F by angle pi/4 around point W. Line Y is the angle bisector of angle ZKF. Point E is the reflection of point R across segment FH. Circle A is the incircle of triangle KFH. Let G be the reflection of circle A across line Y. Point D is the reflection of point Z across line Y. Let I be the inversion of point H in circle G. Find the measure of angle EDI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let LU = 6. Let O be the rotation of point U by angle 7pi/12 around point L. Let P be the orthocenter of triangle LOU. Let Y be the reflection of point L across point P. Point J is the reflection of point Y across segment LO. Find the distance between points J and L. Express your answer as a decimal number rounded to 2 decimal places.

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QR = 10. Let N be the reflection of point R across point Q. Let D be the rotation of point Q by angle pi/4 around point R. Point O is the reflection of point D across point N. Point X is the midpoint of segment QN. Let G be the midpoint of segment XD. What is the distance between points G and O? Express your answer as a decimal number rounded to 2 decimal places.

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DN = 5. Let W be the reflection of point D across point N. Let G be the angle bisector of angle DNW. Point F lies on line G at distance 3 from point D. Let K be the incenter of triangle WFD. Point T is the midpoint of segment WF. Point M is the midpoint of segment TK. What is the length of segment KM? Express your answer as a decimal number rounded to 2 decimal places.

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Let QK = 7. Point S is the midpoint of segment KQ. Let L be the perpendicular bisector of segment SQ. Let F be the reflection of point S across point Q. I is constructed so that SI = 12. Point P lies on line L. Let N be the reflection of point S across point I. Let X be the midpoint of segment KP. Point R is the midpoint of segment PF. What is the length of segment XR? Express your answer as a decimal number rounded to 2 decimal places.

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Let KZ = 7. Let V be the rotation of point Z by angle 5pi/12 around point K. Let X be the rotation of point K by angle 11pi/12 around point Z. Point N is the reflection of point V across segment KZ. Line T passes through point Z perpendicular to segment KZ. Point O is the reflection of point X across line T. Find the circumradius of triangle ONZ. Express your answer as a decimal number rounded to 2 decimal places.

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Let SC = 10. Point F is obtained by rotating point C by angle 2pi/3 about point S. Line V is the angle bisector of angle FCS. Line Z is the reflection of line V across point F. Point P is the reflection of point F across point C. Let Q be the orthocenter of triangle PSF. Point U is the reflection of point F across line Z. What is the length of segment QU? Express your answer as a decimal number rounded to 2 decimal places.

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Let NW = 8. Let K be the midpoint of segment NW. Let M be the perpendicular bisector of segment KN. Point Y lies on line M at distance 2 from point W. Point E is the midpoint of segment KY. Let A be the orthocenter of triangle WKY. Find the inradius of triangle EWA. Express your answer as a decimal number rounded to 2 decimal places.

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RQ = 10. Point D is obtained by rotating point R by angle 5pi/6 about point Q. Let L be the incenter of triangle DRQ. Let F be the circumcircle of triangle DRQ. Point X is the circumcenter of triangle DRQ. Point U is the circumcenter of triangle RLX. Let Y be the inversion of point U in circle F. What is the distance between points Y and X? Express your answer as a decimal number rounded to 2 decimal places.

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Let AG = 10. Line F passes through points A and G. Point N is obtained by rotating point G by angle pi/3 about point A. Let W be the midpoint of segment NG. Let D be the midpoint of segment WA. Point L is the reflection of point W across line F. Let T be the circumcenter of triangle LDN. Point H is the intersection of line F and segment LD. Find the distance between points H and T. Express your answer as a decimal number rounded to 2 decimal places.

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Let IP = 12. Let R be the midpoint of segment PI. Point A is obtained by rotating point P by angle 11pi/12 about point I. Point U is the midpoint of segment AI. Let M be the circumcenter of triangle RAI. Point H lies on segment AI. Let N be the angle bisector of angle HIM. Point G is the reflection of point U across line N. Point S is the midpoint of segment GP. Point E is the midpoint of segment SG. What is the length of segment SE? Express your answer as a decimal number rounded to 2 decimal places.

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Let HB = 2. Let Y be the rotation of point B by angle pi/12 around point H. Line V is the perpendicular bisector of segment HB. Line C passes through point Y perpendicular to segment HB. Point W is obtained by rotating point Y by angle 5pi/12 about point B. Point O is obtained by rotating point W by angle pi/4 about point Y. Point S is the reflection of point O across line V. Segments SY and HB intersect at point F. Point J is the reflection of point S across line C. Find the length of segment JF. Express your answer as a decimal number rounded to 2 decimal places.

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Let SI = 8. E is constructed so that IE = 1. Let Z be the midpoint of segment SE. Point F is the reflection of point Z across point I. Line J passes through point I perpendicular to segment EI. Let Y be the midpoint of segment EI. Point L is the reflection of point Y across line J. What is the distance between points L and F? Express your answer as a decimal number rounded to 2 decimal places.

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US = 2. Point B is obtained by rotating point U by angle 5pi/12 about point S. Point N is the midpoint of segment BS. Let Q be the midpoint of segment UN. Let P be the orthocenter of triangle SUN. Point W is the centroid of triangle SUN. Find the measure of angle QWP, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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WQ = 1. Point L is obtained by rotating point Q by angle pi about point W. Let Y be the perpendicular bisector of segment WL. Point J is obtained by rotating point W by angle 5pi/6 about point Q. Point Z is the reflection of point J across line Y. Point M is the orthocenter of triangle ZJQ. Let H be the midpoint of segment JQ. What is the distance between points M and H? Express your answer as a decimal number rounded to 2 decimal places.

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NB = 7. Let E be the reflection of point B across point N. Point R is obtained by rotating point N by angle pi/12 about point E. Let I be the rotation of point R by angle pi around point E. Let V be the midpoint of segment RN. Let U be the incenter of triangle NBI. Find the circumradius of triangle UVR. Express your answer as a decimal number rounded to 2 decimal places.

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JE = 3. Let F be the midpoint of segment EJ. Let V be the rotation of point J by angle pi/2 around point F. Let D be the line through point J perpendicular to segment EV. Point X is the midpoint of segment FE. Point Z is the reflection of point E across line D. Point G is the reflection of point X across segment EV. Point I is the reflection of point F across line D. Find the measure of angle IZG, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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LE = 3. Point Q is obtained by rotating point E by angle 3pi/4 about point L. Let U be the reflection of point Q across point E. Point J is the reflection of point U across segment LE. Let X be the midpoint of segment LE. What is the length of segment XJ? Express your answer as a decimal number rounded to 2 decimal places.

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CO = 3. Point K is obtained by rotating point O by angle pi/6 about point C. Let L be the rotation of point C by angle pi/3 around point O. Point N is the midpoint of segment KO. Let H be the midpoint of segment OL. Point G is the reflection of point N across point C. Find the inradius of triangle GKH. Express your answer as a decimal number rounded to 2 decimal places.

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Let FX = 1. Let L be the rotation of point X by angle pi/2 around point F. Point P is obtained by rotating point F by angle 11pi/12 about point X. Circle G is the incircle of triangle PXL. Let C be the orthocenter of triangle PXL. Let V be the circumcenter of triangle PXL. Let W be the inversion of point L in circle G. Find the measure of angle CVW, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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CL = 10. Let H be the line through points L and C. Point M lies on line H at distance 4 from point L. Point B is obtained by rotating point L by angle pi/12 about point M. Point N is the reflection of point B across line H. Let Q be the reflection of point L across point B. Line R passes through points M and B. Line Z is the perpendicular bisector of segment LN. Let V be the midpoint of segment BQ. Let G be the reflection of line Z across line R. Point D is the reflection of point Q across line G. Let U be the midpoint of segment VQ. Find the circumradius of triangle DNU. Express your answer as a decimal number rounded to 2 decimal places.

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GT = 4. J is constructed so that TJ = 5. Let U be the reflection of point J across point T. Let X be the midpoint of segment GU. Let O be the midpoint of segment JG. What is the distance between points O and X? Express your answer as a decimal number rounded to 2 decimal places.

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SL = 3. Point I is obtained by rotating point L by angle 3pi/4 about point S. Point V is the midpoint of segment LI. Let Q be the midpoint of segment VL. Let N be the line through points L and S. Point Z is the midpoint of segment QS. Point X is the orthocenter of triangle QZI. Point P is the reflection of point V across line N. Point A is the reflection of point X across line N. Find the length of segment PA. Express your answer as a decimal number rounded to 2 decimal places.

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EK = 9. Point Y is obtained by rotating point K by angle pi/4 about point E. Point T is the midpoint of segment YK. Point B is the reflection of point K across point Y. Point N is the midpoint of segment BT. Point X is the reflection of point T across segment NE. Point U is the incenter of triangle YNE. Point W is the midpoint of segment XT. What is the distance between points W and U? Express your answer as a decimal number rounded to 2 decimal places.

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Let LV = 3. Point G is obtained by rotating point L by angle 5pi/6 about point V. Line E is the perpendicular bisector of segment GL. Line Y is the reflection of line E across point G. Line S is the reflection of line Y across line E. Let A be the midpoint of segment LV. Point M is the circumcenter of triangle VLG. Let D be the incenter of triangle VLG. Point B is the reflection of point M across line S. Find the circumradius of triangle ABD. Express your answer as a decimal number rounded to 2 decimal places.

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RX = 9. Point W is obtained by rotating point R by angle 11pi/12 about point X. Point N is the midpoint of segment XR. Let Z be the midpoint of segment WR. Point H is obtained by rotating point W by angle 5pi/12 about point Z. Let Y be the reflection of point W across point R. Let B be the angle bisector of angle YZN. Point K is the reflection of point X across line B. What is the circumradius of triangle KRH? Express your answer as a decimal number rounded to 2 decimal places.

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UF = 4. Point D is the midpoint of segment FU. Point T is obtained by rotating point U by angle 11pi/12 about point F. Let Q be the midpoint of segment FT. Point C is the reflection of point U across segment TD. Line J is the angle bisector of angle CQU. Point I is the reflection of point U across line J. What is the measure of angle CDI, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let LN = 7. Let B be the rotation of point L by angle 7pi/12 around point N. Point Q is the reflection of point N across segment BL. Point G is the reflection of point L across point B. Point S is the reflection of point Q across point B. What is the inradius of triangle GSN? Express your answer as a decimal number rounded to 2 decimal places.

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Let WL = 4. Let T be the reflection of point L across point W. Let R be the rotation of point W by angle 11pi/12 around point L. Point Z is the orthocenter of triangle TWR. Point F is the midpoint of segment RW. Point J is the orthocenter of triangle RLZ. Find the length of segment FJ. Express your answer as a decimal number rounded to 2 decimal places.

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RS = 10. Let D be the midpoint of segment SR. Point Y is obtained by rotating point S by angle pi/12 about point R. Let Q be the line through points Y and D. Point E is the reflection of point S across line Q. Let T be the midpoint of segment YE. Find the length of segment TD. Express your answer as a decimal number rounded to 2 decimal places.

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Let KU = 3. Point H is the reflection of point U across point K. Let R be the line through points K and U. Let G be the rotation of point U by angle 2pi/3 around point H. Let F be the line through point G parallel to line R. Let V be the reflection of point G across point U. Let L be the rotation of point G by angle pi/12 around point V. Point S is the reflection of point H across line F. Let E be the reflection of point H across point S. Point X is the reflection of point H across point U. What is the circumradius of triangle XEL? Express your answer as a decimal number rounded to 2 decimal places.

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ZO = 7. Let B be the midpoint of segment ZO. Let V be the rotation of point B by angle pi/4 around point O. Let X be the midpoint of segment BZ. Let J be the midpoint of segment ZX. Line W passes through point V perpendicular to segment XZ. Point L is the reflection of point J across line W. Let T be the circumcircle of triangle VXZ. Let G be the inversion of point L in circle T. Point S is the midpoint of segment GJ. What is the length of segment SJ? Express your answer as a decimal number rounded to 2 decimal places.

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SN = 8. Let X be the midpoint of segment NS. Let K be the rotation of point N by angle pi/12 around point X. Let H be the line through points K and S. Line T passes through point N parallel to line H. Point Q is the reflection of point N across point K. Point L is the reflection of point Q across line T. Find the inradius of triangle LQS. Express your answer as a decimal number rounded to 2 decimal places.

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OP = 5. Let Y be the line through points O and P. Let H be the perpendicular bisector of segment OP. Let J be the midpoint of segment PO. Point X is the reflection of point J across point P. Line D is the reflection of line H across point P. Line K passes through point O perpendicular to line Y. Point S lies on line K at distance 9 from point J. Point B is the reflection of point S across line D. Let W be the midpoint of segment XS. What is the circumradius of triangle OWB? Express your answer as a decimal number rounded to 2 decimal places.

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DT = 2. Let C be the rotation of point D by angle 3pi/4 around point T. Let Y be the circumcenter of triangle TCD. Point R is the incenter of triangle TCD. Let Z be the line through point Y perpendicular to segment TC. Point Q is the reflection of point T across line Z. What is the length of segment RQ? Express your answer as a decimal number rounded to 2 decimal places.

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Let EP = 6. Line V is the perpendicular bisector of segment PE. Point L lies on line V. Point F is obtained by rotating point E by angle 11pi/12 about point L. Let W be the midpoint of segment EP. Let G be the midpoint of segment FP. Let Z be the midpoint of segment PL. Find the measure of angle GZW, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let CN = 4. Let E be the rotation of point N by angle 5pi/12 around point C. Let L be the reflection of point C across point N. Point U is the midpoint of segment CE. Let F be the orthocenter of triangle LEN. Let K be the reflection of point N across point F. Point H is the circumcenter of triangle LEN. Let M be the incenter of triangle LEN. Let Z be the midpoint of segment KU. What is the length of segment ZM? Express your answer as a decimal number rounded to 2 decimal places.

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Let SM = 4. Line X is the perpendicular bisector of segment MS. Point F is obtained by rotating point S by angle pi/4 about point M. Point R is the reflection of point F across line X. Let J be the angle bisector of angle FRM. Point C is the reflection of point R across point M. Point H is the reflection of point C across point R. Point I is the reflection of point C across line J. Find the measure of angle IHS, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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RL = 9. Line U is the perpendicular bisector of segment LR. Let E be the line through point L parallel to line U. Let C be the midpoint of segment RL. Point P is obtained by rotating point L by angle 3pi/4 about point R. Line K passes through point P parallel to segment LR. Point A is the reflection of point C across line E. Point Q is the reflection of point C across line K. What is the circumradius of triangle QPA? Express your answer as a decimal number rounded to 2 decimal places.

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Let MG = 3. Let E be the rotation of point M by angle 5pi/6 around point G. Point V is the midpoint of segment EG. Let D be the incenter of triangle EGM. Let S be the centroid of triangle EGM. What is the measure of angle VDS, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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SG = 12. Point L is obtained by rotating point S by angle pi/2 about point G. Point Z is obtained by rotating point S by angle pi/2 about point L. Point A is the circumcenter of triangle ZSL. Line R is the angle bisector of angle ZAG. Point U is the reflection of point S across line R. What is the measure of angle USG, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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EJ = 6. Let C be the rotation of point J by angle 5pi/12 around point E. Point G is the reflection of point C across segment EJ. Point Y is the reflection of point J across point G. Point Z is the incenter of triangle EJC. Let H be the midpoint of segment JC. What is the measure of angle ZHY, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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SB = 9. Let R be the rotation of point S by angle 7pi/12 around point B. Let Z be the reflection of point B across point R. Point D is the centroid of triangle BSR. Point M is the orthocenter of triangle BSR. Find the measure of angle ZDM, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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SO = 10. Let T be the rotation of point O by angle 2pi/3 around point S. Let R be the reflection of point T across point O. Let U be the reflection of point S across point T. Let I be the reflection of point S across point O. What is the circumradius of triangle URI? Express your answer as a decimal number rounded to 2 decimal places.

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JG = 2. Let I be the rotation of point G by angle 7pi/12 around point J. Point N is the midpoint of segment GI. Point R is the centroid of triangle GIJ. Point Y is the reflection of point R across segment GI. Point B is the reflection of point N across segment JG. What is the distance between points Y and B? Express your answer as a decimal number rounded to 2 decimal places.

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Let XG = 6. Point B is obtained by rotating point X by angle pi about point G. Let Z be the rotation of point B by angle 7pi/12 around point G. Let U be the midpoint of segment ZG. Point V is on segment UX. Let W be the midpoint of segment UB. What is the measure of angle VWG, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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VD = 11. Point U is obtained by rotating point V by angle 3pi/4 about point D. Line H is the angle bisector of angle VUD. Let R be the midpoint of segment VD. Point A is the reflection of point R across segment DU. Point P is the reflection of point V across line H. Find the distance between points A and P. Express your answer as a decimal number rounded to 2 decimal places.

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NO = 1. Point Y is obtained by rotating point O by angle pi/12 about point N. Let R be the midpoint of segment ON. Point E is the reflection of point R across point N. Let K be the orthocenter of triangle YRE. Find the distance between points K and Y. Express your answer as a decimal number rounded to 2 decimal places.

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VU = 8. Let J be the rotation of point V by angle pi/4 around point U. Let B be the midpoint of segment VU. Point T is the orthocenter of triangle BVJ. Let K be the midpoint of segment VJ. Find the measure of angle UTK, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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HQ = 6. Let Y be the midpoint of segment QH. Point A is the reflection of point Y across point H. Point B is obtained by rotating point A by angle 3pi/4 about point Y. Let K be the midpoint of segment BH. Let R be the line through point K perpendicular to segment AB. Point V is the reflection of point B across line R. What is the length of segment KV? Express your answer as a decimal number rounded to 2 decimal places.

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Let RY = 8. Let H be the vector from point R to point Y. Point O is obtained by rotating point Y by angle 3pi/4 about point R. Let L be point O translated by vector H. Point K is the midpoint of segment RY. Let D be the reflection of point K across point Y. Point M is the reflection of point K across segment RO. Point C is the reflection of point D across segment RO. Point E is the reflection of point D across segment MO. What is the inradius of triangle ELY? Express your answer as a decimal number rounded to 2 decimal places.

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Let BN = 12. Let G be the rotation of point B by angle 2pi/3 around point N. Point W is the reflection of point B across point N. Let I be the midpoint of segment NW. Line V is the perpendicular bisector of segment NG. Let K be the midpoint of segment IW. Point L is the reflection of point K across line V. Find the circumradius of triangle GNL. Express your answer as a decimal number rounded to 2 decimal places.

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Let RP = 9. Let S be the rotation of point P by angle 11pi/12 around point R. Let N be the perpendicular bisector of segment SP. Point O is the reflection of point S across point R. Line I is the perpendicular bisector of segment SR. Let T be the reflection of line I across line N. Point Z is the midpoint of segment SR. Line T and segment SO intersect at point H. 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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Let TL = 12. Point P is obtained by rotating point T by angle pi/4 about point L. Point M is the orthocenter of triangle TPL. Circle V is the incircle of triangle TPL. Point Y is the reflection of point M across segment PL. Let K be the inversion of point Y in circle V. Find the length of segment PK. Express your answer as a decimal number rounded to 2 decimal places.

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Let OH = 2. Let U be the rotation of point H by angle pi/6 around point O. Point V is obtained by rotating point H by angle 2pi/3 about point U. Point T is the midpoint of segment OV. Let K be the line through point U perpendicular to segment HO. Point R is the reflection of point V across line K. Let J be the reflection of point H across point V. What is the circumradius of triangle HJR? Express your answer as a decimal number rounded to 2 decimal places.

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DL = 10. Point B is the midpoint of segment LD. Let M be the perpendicular bisector of segment DL. Point R is on line M with |RD| = 4. Line X passes through point R perpendicular to segment DR. Point S is the midpoint of segment DR. Point C is the reflection of point B across line X. Find the length of segment CS. Express your answer as a decimal number rounded to 2 decimal places.

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EV = 2. Point G is the reflection of point E across point V. Let P be the reflection of point E across point G. Point Y is obtained by rotating point G by angle 7pi/12 about point V. Point Q is obtained by rotating point P by angle 5pi/12 about point Y. Point J is the midpoint of segment YE. Let W be the reflection of point P across point J. Let U be the line through point W parallel to segment YE. Point O is the reflection of point Q across line U. What is the measure of angle JGO, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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TA = 12. Point S is obtained by rotating point T by angle 5pi/12 about point A. Let X be the reflection of point T across point A. Point U is the midpoint of segment AS. Point I is the midpoint of segment UX. What is the measure of angle STI, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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OF = 5. Point Z is obtained by rotating point O by angle 2pi/3 about point F. Point Y is the incenter of triangle OZF. Let I be the centroid of triangle OZF. Let S be the incircle of triangle OZF. Circle B is the reflection of circle S across point I. Let D be the inversion of point I in circle B. Point W is the reflection of point Y across segment ZF. Find the distance between points W and D. Express your answer as a decimal number rounded to 2 decimal places.

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UB = 3. Let E be the rotation of point B by angle 5pi/12 around point U. Let Z be the vector from point B to point U. Let H be point U translated by vector Z. Point O is the midpoint of segment EH. Point P is the reflection of point U across point H. Let Q be the midpoint of segment PE. Find the circumradius of triangle OQP. Express your answer as a decimal number rounded to 2 decimal places.

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NW = 8. Point Y is obtained by rotating point W by angle 3pi/4 about point N. Line P passes through point N perpendicular to segment NY. Let Q be the reflection of point Y across point N. Let X be the reflection of point Q across point W. Point D is the reflection of point X across line P. Point M is the centroid of triangle YWN. What is the distance between points M and D? Express your answer as a decimal number rounded to 2 decimal places.

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Let IS = 11. Let Z be the rotation of point S by angle 3pi/4 around point I. Point F is obtained by rotating point Z by angle 2pi/3 about point S. Let T be the incenter of triangle ZFI. Point L is the reflection of point S across point I. Let A be the circumcenter of triangle ZFI. Find the measure of angle ATL, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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ZC = 8. Let U be the reflection of point Z across point C. Point K is obtained by rotating point U by angle 11pi/12 about point Z. Point H is the reflection of point K across segment ZU. Let F be the centroid of triangle ZUK. Point X is the reflection of point Z across point H. Find the distance between points F and X. Express your answer as a decimal number rounded to 2 decimal places.

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Let CJ = 8. Line P is the perpendicular bisector of segment JC. Let H be the rotation of point C by angle 11pi/12 around point J. Point M is the reflection of point H across line P. Let X be the centroid of triangle CMH. Point E is the reflection of point X across line P. Let L be the midpoint of segment HC. What is the distance between points E and L? Express your answer as a decimal number rounded to 2 decimal places.

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QJ = 2. Let W be the rotation of point J by angle pi/4 around point Q. Let U be the rotation of point W by angle 2pi/3 around point Q. Let C be the rotation of point Q by angle 11pi/12 around point U. Point I is the midpoint of segment CU. Line A is the perpendicular bisector of segment CI. Let H be the circumcenter of triangle JWQ. Point K is the reflection of point H across line A. Find the inradius of triangle QHK. Express your answer as a decimal number rounded to 2 decimal places.

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Let UJ = 4. Point P is the reflection of point U across point J. Let F be the rotation of point J by angle pi/3 around point U. Point E is the midpoint of segment JF. Let I be the midpoint of segment EP. Point Y is the circumcenter of triangle EUF. Point H is the midpoint of segment UY. Find the measure of angle PHI, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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LZ = 12. Let B be the midpoint of segment ZL. Let E be the rotation of point L by angle 7pi/12 around point Z. Point P is the reflection of point E across point Z. Let I be the midpoint of segment PZ. Let F be the orthocenter of triangle PIB. Find the distance between points F and L. Express your answer as a decimal number rounded to 2 decimal places.

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Let VD = 5. Let K be the rotation of point V by angle pi/4 around point D. Let G be the rotation of point K by angle 2pi/3 around point V. Line W is the perpendicular bisector of segment DV. Let X be the reflection of line W through point D. Point P is the reflection of point V across point K. Point E is the reflection of point G across line X. Let S be the reflection of point P across point E. What is the inradius of triangle DES? Express your answer as a decimal number rounded to 2 decimal places.

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Let IB = 1. Let L be the vector from point B to point I. Let G be the rotation of point I by angle pi/3 around point B. Let Z be the midpoint of segment BG. Point M is the translation of point I by vector L. Line R passes through point M perpendicular to segment MZ. Point N is the midpoint of segment MZ. Point A is the reflection of point G across line R. Point C is the reflection of point M across segment NG. What is the length of segment MC? Express your answer as a decimal number rounded to 2 decimal places.

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YB = 12. Line T is the perpendicular bisector of segment YB. Line Q is the reflection of line T across point B. Point H is on line Q with |HB| = 3. Let L be the orthocenter of triangle YHB. Point O is the reflection of point L across segment YH. Point V is the reflection of point Y across line Q. Point W is the reflection of point O across point V. Find the length of segment YW. Express your answer as a decimal number rounded to 2 decimal places.

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ZQ = 5. Point D is obtained by rotating point Q by angle pi/6 about point Z. Point S is the incenter of triangle QDZ. Point V is the orthocenter of triangle QDZ. Point A is the reflection of point V across segment DZ. Find the length of segment SA. Express your answer as a decimal number rounded to 2 decimal places.

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Let BG = 4. K is constructed so that GK = 6. Let P be the midpoint of segment KB. Let N be the midpoint of segment KP. Point A is the midpoint of segment PG. Find the length of segment AN. Express your answer as a decimal number rounded to 2 decimal places.

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WK = 3. Let Q be the rotation of point W by angle 11pi/12 around point K. Let P be the reflection of point K across point Q. Point L is the midpoint of segment WP. Point E is the reflection of point K across point L. Point F is the reflection of point L across segment KW. What is the inradius of triangle EWF? Express your answer as a decimal number rounded to 2 decimal places.

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XY = 9. Let G be the rotation of point X by angle pi/2 around point Y. Let W be the line through point Y perpendicular to segment XY. Point R is on line W with |RY| = 8. Let Z be the midpoint of segment YG. Let J be the midpoint of segment XY. Point V is the reflection of point G across point J. Point E is the midpoint of segment RX. Let Q be the midpoint of segment VX. What is the measure of angle ZEQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let RP = 7. Let C be the vector from point P to point R. Point I is obtained by rotating point R by angle 11pi/12 about point P. Let Y be point I translated by vector C. Let N be the midpoint of segment IY. Let X be the midpoint of segment RI. Point B is the incenter of triangle YRX. Find the circumradius of triangle IBN. Express your answer as a decimal number rounded to 2 decimal places.

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IF = 2. Point O is the reflection of point I across point F. Point A is obtained by rotating point I by angle pi/6 about point F. Let C be the midpoint of segment IF. Point J is the midpoint of segment CO. Let V be the reflection of point C across point A. Point L is the midpoint of segment JV. What is the distance between points L and J? Express your answer as a decimal number rounded to 2 decimal places.

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Let ZQ = 1. Let D be the rotation of point Q by angle pi/4 around point Z. Point E is the reflection of point Q across point Z. Point K is the midpoint of segment QZ. Let Y be the incircle of triangle KZD. Let O be the inversion of point Q in circle Y. Point J is the incenter of triangle KZD. What is the measure of angle OJE, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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TB = 1. Let U be the rotation of point B by angle pi/2 around point T. Point O is the incenter of triangle UTB. Let K be the line through points T and O. Point N is the intersection of segment BU and line K. Point M is the midpoint of segment BO. What is the length of segment MN? Express your answer as a decimal number rounded to 2 decimal places.

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SY = 10. Let X be the rotation of point S by angle pi/2 around point Y. Let N be the midpoint of segment XY. Line A is the perpendicular bisector of segment NX. Let B be the reflection of point X across point S. Line J is the perpendicular bisector of segment SN. Point D is the reflection of point B across line J. Point K is the reflection of point D across line A. Let T be the reflection of point N across point K. Find the inradius of triangle KTD. Express your answer as a decimal number rounded to 2 decimal places.

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Let EA = 11. Let T be the reflection of point A across point E. Let K be the rotation of point A by angle 11pi/12 around point E. Point D is the orthocenter of triangle EKA. Point P is the midpoint of segment TK. Point M is the reflection of point T across point A. Let R be the circumcenter of triangle MPD. Find the measure of angle RAD, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let QN = 6. Point C is the reflection of point Q across point N. Let K be the reflection of point N across point C. Line Z passes through points Q and C. Let S be the line through point C perpendicular to line Z. Let M be the rotation of point K by angle pi/2 around point N. Let V be the reflection of point M across point Q. Let I be the angle bisector of angle QVK. Point G is obtained by rotating point V by angle 7pi/12 about point Q. Line U is the reflection of line S across line I. Line D is the reflection of line I across point C. Let P be the midpoint of segment CG. Lines D and U intersect at point J. What is the measure of angle JQP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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HO = 7. Point S is the reflection of point H across point O. Let G be the midpoint of segment OH. Let U be the rotation of point G by angle pi/4 around point S. Point X is the reflection of point G across point U. Let M be the midpoint of segment OU. Point Y is the midpoint of segment XM. What is the measure of angle YUS, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let FI = 2. Let D be the rotation of point I by angle pi/4 around point F. Point O is the midpoint of segment FD. Let Y be the reflection of point O across point F. Point N is the midpoint of segment YF. Point E is the midpoint of segment IF. Point U is the reflection of point D across segment IN. What is the measure of angle YFU, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let UH = 10. Let Z be the midpoint of segment UH. Point T is obtained by rotating point U by angle pi/6 about point Z. Point G is the reflection of point T across point H. Point J is the orthocenter of triangle ZUT. What is the inradius of triangle JHG? Express your answer as a decimal number rounded to 2 decimal places.

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Let PZ = 12. Line X is the perpendicular bisector of segment ZP. Point L is the reflection of point Z across point P. Let O be the rotation of point P by angle pi/3 around point L. Point A is the reflection of point O across line X. Let N be the centroid of triangle LAO. Point T is the orthocenter of triangle LAO. What is the distance between points N and T? Express your answer as a decimal number rounded to 2 decimal places.

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Let QR = 12. O is placed 12 units away from R. Let I be the rotation of point R by angle 5pi/6 around point O. Point M is the midpoint of segment OR. Let C be the midpoint of segment RQ. Let W be the reflection of point M across point C. Point E is the reflection of point R across segment WI. What is the length of segment IE? Express your answer as a decimal number rounded to 2 decimal places.

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SJ = 5. Let P be the rotation of point S by angle 11pi/12 around point J. Let L be the perpendicular bisector of segment PJ. Point H is the reflection of point P across point S. Point G is the orthocenter of triangle HJP. Point N is the reflection of point H across line L. What is the length of segment NG? Express your answer as a decimal number rounded to 2 decimal places.

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HS = 6. Let X be the rotation of point H by angle pi/12 around point S. Let A be the midpoint of segment XH. Point P is obtained by rotating point X by angle pi/2 about point S. Let K be the reflection of point A across point H. Let O be the midpoint of segment PA. Let U be the reflection of point S across point A. Find the circumradius of triangle KOU. Express your answer as a decimal number rounded to 2 decimal places.

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Let NS = 10. O is constructed so that SO = 5. Let A be the centroid of triangle SON. Let M be the line through point A parallel to segment NS. Point J is the intersection of line M and segment ON. What is the distance between points J and A? Express your answer as a decimal number rounded to 2 decimal places.

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Let HY = 8. Let I be the midpoint of segment YH. Let A be the reflection of point I across point H. Let Z be the midpoint of segment YA. Line R is the perpendicular bisector of segment IH. Let M be the midpoint of segment YZ. Let V be the line through point M parallel to line R. Let C be the line through point A parallel to line R. Let U be the rotation of point H by angle pi/3 around point Z. Point G is the reflection of point A across point U. Let F be the midpoint of segment GU. Line P is the angle bisector of angle FMZ. Let L be the reflection of line P across line C. Point S is the intersection of lines L and V. Find the measure of angle HSF, 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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JZ = 9. Let C be the reflection of point J across point Z. Let Y be the rotation of point Z by angle pi/12 around point C. Let F be the midpoint of segment JY. Let H be the incircle of triangle JYC. Let I be the incenter of triangle JYC. Point T is the inversion of point Y with respect to circle H. Point P is the reflection of point T across point F. Find the measure of angle CIP, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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MK = 8. Point A is obtained by rotating point K by angle pi/4 about point M. Point N is the midpoint of segment MK. Point O is the reflection of point N across segment MA. Let Y be the midpoint of segment AK. What is the circumradius of triangle ONY? 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 BE = 3. Let L be the reflection of point E across point B. Let A be the rotation of point E by angle 11pi/12 around point L. Let Z be the midpoint of segment AB. Point K is the reflection of point B across point A. Point F is the reflection of point L across point E. What is the inradius of triangle FKZ? 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 = 8. Let R be the reflection of point S across point I. Line B passes through points R and S. Point L is obtained by rotating point S by angle pi/12 about point R. Point Q is the reflection of point L across line B. X is constructed so that RX = 3. Point V is the midpoint of segment IQ. Point J is the reflection of point V across segment LR. Point E is the midpoint of segment LR. Find the distance between points E and J. Express your answer as a decimal number rounded to 2 decimal places.

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Let PC = 8. Let W be the vector from point P to point C. Let I be the rotation of point P by angle 5pi/12 around point C. Point N is the translation of point I by vector W. Point D is the incenter of triangle NCI. Line U is the perpendicular bisector of segment IN. Line J is the reflection of line U across point C. Line J and segment IN intersect at point Q. Point E is the reflection of point D across segment NC. Find the length of segment QE. Express your answer as a decimal number rounded to 2 decimal places.

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CT = 4. Point S is obtained by rotating point T by angle pi/6 about point C. Point G is the reflection of point C across segment ST. Point E is the reflection of point G across point T. Point L is the reflection of point G across point E. What is the circumradius of triangle ELS? Express your answer as a decimal number rounded to 2 decimal places.

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Let VC = 10. Point Q is obtained by rotating point V by angle pi/4 about point C. Point W is the reflection of point Q across point V. Point I is the midpoint of segment CQ. Let M be the midpoint of segment WC. Point G is the reflection of point M across segment CQ. Find the measure of angle IMG, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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NK = 10. Point E is obtained by rotating point K by angle pi/6 about point N. Let L be the centroid of triangle NEK. Let Q be the orthocenter of triangle NEK. Let C be the midpoint of segment QK. What is the length of segment CL? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let PB = 8. Let O be the rotation of point P by angle pi/4 around point B. Point S is the midpoint of segment BP. Point K is the midpoint of segment SP. Point G is the reflection of point K across point P. What is the circumradius of triangle KGO? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MJ = 12. Let B be the midpoint of segment MJ. Let Q be the reflection of point B across point M. Line I is the perpendicular bisector of segment BJ. Let E be the reflection of point M across point Q. Let T be the rotation of point M by angle 2pi/3 around point E. Point W is the midpoint of segment TB. Point Y is the reflection of point M across line I. Point O is the centroid of triangle TWE. Find the distance between points Y 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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GU = 3. Line X is the perpendicular bisector of segment UG. Let Z be the reflection of line X through point U. Point B is the reflection of point U across point G. Let Y be the rotation of point U by angle pi/3 around point B. Let R be the midpoint of segment UG. Point P is the midpoint of segment YR. Let T be the reflection of line Z through point P. Point D is the reflection of point P across line T. Find the measure of angle BDP, 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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EH = 2. Point R is the midpoint of segment HE. Point T is the reflection of point R across point H. Let B be the reflection of point H across point T. Point L is obtained by rotating point B by angle 3pi/4 about point R. Let P be the rotation of point T by angle pi/6 around point E. Line D is the perpendicular bisector of segment RT. Point X is the reflection of point P across line D. Point K is the reflection of point R across segment XL. Find the length of segment XK. Express your answer as a decimal number rounded to 2 decimal places.

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Let IE = 9. Let U be the reflection of point I across point E. Let T be the rotation of point U by angle 2pi/3 around point I. Point C is the midpoint of segment ET. Let W be the midpoint of segment TC. Let Q be the incenter of triangle ETU. Line G is the perpendicular bisector of segment IQ. Point N is the circumcenter of triangle ETU. Point D is the reflection of point W across line G. Find the inradius of triangle NDT. Express your answer as a decimal number rounded to 2 decimal places.

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Let DC = 9. Point G is obtained by rotating point D by angle 3pi/4 about point C. Let A be the perpendicular bisector of segment CG. Let J be the midpoint of segment DC. Line F passes through point D parallel to line A. Point T is the reflection of point J across line F. Find the distance between points J and T. Express your answer as a decimal number rounded to 2 decimal places.

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Let OC = 3. Point A is the reflection of point C across point O. Let N be the rotation of point C by angle 7pi/12 around point O. Line B is the perpendicular bisector of segment NO. Line V passes through point A parallel to line B. Point J is the reflection of point O across line V. Point U is the reflection of point N across segment OA. Point T is the midpoint of segment OA. Segments OA and JU intersect at point H. Find the distance between points H and T. Express your answer as a decimal number rounded to 2 decimal places.

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MC = 2. Let E be the rotation of point C by angle 7pi/12 around point M. Point O is the centroid of triangle CME. Let V be the incenter of triangle CME. Circle A is the circumcircle of triangle CME. Let G be the inversion of point V in circle A. Let Y be the midpoint of segment OC. Let T be the incircle of triangle CME. Line P passes through point G perpendicular to segment OC. Let W be the reflection of circle T across line P. Let N be the inversion of point V in circle W. Find the circumradius of triangle NYM. Express your answer as a decimal number rounded to 2 decimal places.

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Let SK = 7. Let N be the rotation of point S by angle pi/12 around point K. Point U is the reflection of point S across segment NK. Point T is the reflection of point N across point U. Let O be the midpoint of segment SN. Let Y be the reflection of point O across point K. What is the length of segment TY? Express your answer as a decimal number rounded to 2 decimal places.

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Let VT = 6. Let X be the reflection of point V across point T. Point U is obtained by rotating point T by angle pi/4 about point V. Point G is the midpoint of segment UT. Let Z be the midpoint of segment UG. Let H be the orthocenter of triangle UXZ. Point C is the midpoint of segment ZH. What is the distance between points C and U? 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 WB = 11. Let X be the reflection of point W across point B. Let H be the rotation of point W by angle pi/12 around point X. Let D be the incenter of triangle HWX. Point P is the reflection of point B across segment HD. What is the distance between points P and X? Express your answer as a decimal number rounded to 2 decimal places.

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DH = 1. Let O be the rotation of point H by angle 5pi/6 around point D. Point R is obtained by rotating point O by angle 5pi/12 about point D. Point V is the reflection of point R across point O. Point Y is the reflection of point V across segment DH. Let J be the perpendicular bisector of segment HY. Let Z be the reflection of point H across point V. Point X is the reflection of point O across line J. What is the measure of angle XVZ, 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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CK = 1. Let I be the line through points K and C. Let Q be the line through point K perpendicular to line I. Point V is the reflection of point K across point C. Point G is on line Q with |GC| = 5. Point U is the reflection of point V across point G. Point M is the midpoint of segment GC. Let J be the midpoint of segment UK. Find the circumradius of triangle JMC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let DH = 11. Let T be the reflection of point D across point H. Point Y is obtained by rotating point D by angle 3pi/4 about point T. Let P be the reflection of point D across point Y. Let V be the line through points H and P. Point C is the reflection of point T across line V. Line K is the angle bisector of angle HTC. Point L is the reflection of point P across line K. Find the measure of angle CLP, 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 KQ = 2. Let L be the rotation of point K by angle 3pi/4 around point Q. Let F be the incenter of triangle LQK. Let O be the circumcenter of triangle LQK. Point H is the midpoint of segment KF. Line X passes through points O and H. Segment LQ and line X intersect at point Z. What is the distance between points F and Z? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BX = 9. Point R is obtained by rotating point X by angle 7pi/12 about point B. Let U be the perpendicular bisector of segment RX. Let W be the reflection of line U through point X. Let G be the line through points R and X. Point E is the midpoint of segment RB. Lines G and W intersect at point J. Find the circumradius of triangle BJE. 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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DK = 1. Point O is the reflection of point D across point K. Let Y be the rotation of point D by angle 5pi/6 around point O. Line I is the perpendicular bisector of segment DY. Point L is the reflection of point K across line I. Point U is the reflection of point L across point D. What is the inradius of triangle UYL? 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 DU = 12. Let J be the line through point D perpendicular to segment UD. Point Z is obtained by rotating point D by angle pi/4 about point U. Point Y is the midpoint of segment ZU. Let I be the rotation of point D by angle pi/6 around point U. Let G be the angle bisector of angle UYD. Point B is the midpoint of segment IZ. Point X is the reflection of point U across line G. Point C is the reflection of point X across line J. Find the circumradius of triangle BCU. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let PU = 2. Point S is obtained by rotating point P by angle pi/12 about point U. Point Z is the midpoint of segment SP. Let A be the line through point P perpendicular to segment US. Point M is the reflection of point Z across line A. Let R be the reflection of point Z across point M. Let Y be the reflection of point U across point R. Point Q is the midpoint of segment US. What is the length of segment QY? Express your answer as a decimal number rounded to 2 decimal places.

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


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HR = 8. Point A is obtained by rotating point R by angle 5pi/12 about point H. Let B be the midpoint of segment RH. Point L is the reflection of point B across point R. Let J be the reflection of point L across point B. Find the distance between points J and B. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
ZJ = 9. Let X be the rotation of point Z by angle 2pi/3 around point J. Point V is obtained by rotating point J by angle 5pi/6 about point Z. Point C is the reflection of point X across segment ZJ. Let T be the reflection of point Z across point J. Let N be the midpoint of segment ZC. Find the circumradius of triangle TNV. 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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ME = 4. Point L is obtained by rotating point E by angle pi/2 about point M. Let A be the midpoint of segment LM. Let N be the line through points M and L. Point D is on line N with |DL| = 7. Point Z is the reflection of point D across segment ME. Let Y be the reflection of point L across point D. Let P be the midpoint of segment YA. Let X be the midpoint of segment EZ. Let S be the midpoint of segment XP. Point I is the reflection of point S across line N. What is the measure of angle IXM, 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 DJ = 1. Point V is the reflection of point J across point D. Point Z is obtained by rotating point D by angle 3pi/4 about point J. Let Q be the midpoint of segment JZ. Let O be the reflection of point V across point Q. Let I be the incenter of triangle JZD. Find the length of segment IO. Express your answer as a decimal number rounded to 2 decimal places.

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Let HS = 8. Point M is obtained by rotating point S by angle 3pi/4 about point H. Point E is the midpoint of segment SH. Point T is the reflection of point E across point S. Let R be the midpoint of segment ST. Let D be the line through point M parallel to segment HE. Point P is the reflection of point R across line D. What is the inradius of triangle MPE? 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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JT = 1. Let V be the vector from point J to point T. Let K be the perpendicular bisector of segment TJ. Point E is the translation of point T by vector V. Let A be the line through point J parallel to line K. Line Y passes through point J perpendicular to line A. Let L be the line through point E perpendicular to line Y. Point W is obtained by rotating point T by angle 5pi/12 about point J. Point P is the reflection of point W across line L. Point Z is the reflection of point E across line K. Let H be the reflection of point Z across point W. Point X is the midpoint of segment TH. Find the circumradius of triangle TPX. 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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EY = 2. Let M be the perpendicular bisector of segment YE. Point L lies on line M at distance 10 from point Y. Let S be the orthocenter of triangle YLE. Let I be the line through point E perpendicular to segment EY. Point Q is the reflection of point Y across line I. Find the length of segment QS. Express your answer as a decimal number rounded to 2 decimal places.

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KF = 8. Point D is obtained by rotating point K by angle pi/4 about point F. Point E is the midpoint of segment FK. Point S is the reflection of point E across point F. Point Q is the reflection of point S across segment KD. Point H is the midpoint of segment KD. Find the measure of angle EQH, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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BD = 1. Point Q is obtained by rotating point B by angle 2pi/3 about point D. Let Y be the midpoint of segment BD. Point U is the centroid of triangle DQB. Point V is the midpoint of segment UB. Find the distance between points V and Y. Express your answer as a decimal number rounded to 2 decimal places.

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PL = 11. Let F be the rotation of point L by angle 7pi/12 around point P. Let Z be the rotation of point P by angle 2pi/3 around point L. Point C is obtained by rotating point F by angle 3pi/4 about point P. Point S is the reflection of point P across point Z. Point R is the reflection of point S across point P. Let U be the centroid of triangle LZC. What is the measure of angle ZRU, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let QE = 9. Point P is obtained by rotating point E by angle 11pi/12 about point Q. Let N be the midpoint of segment QP. Let W be the centroid of triangle QEN. Let J be the midpoint of segment NW. What is the length of segment JW? Express your answer as a decimal number rounded to 2 decimal places.

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Let FI = 9. Let B be the rotation of point F by angle pi/6 around point I. Let R be the reflection of point I across point F. Point E is obtained by rotating point B by angle pi/2 about point F. Let X be the line through point E parallel to segment BR. Circle G is the incircle of triangle EBR. Let H be the centroid of triangle EBR. Let D be the midpoint of segment FH. Let Q be the reflection of circle G across line X. Let L be the inversion of point B in circle Q. What is the measure of angle BDL, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Point T is the centroid of triangle B. Let U be the midpoint of segment BR. Let V be the reflection of point T across point U. Let J be the midpoint of segment VQ. What is the length of segment UJ? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let XM = 11. Let E be the rotation of point X by angle 5pi/12 around point M. Let T be the incenter of triangle XME. Point Y is the midpoint of segment XT. Point V is the midpoint of segment XY. Point U is the reflection of point V across segment XM. Find the distance between points U and Y. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let YZ = 6. Point Q is the reflection of point Z across point Y. Let N be the rotation of point Q by angle 3pi/4 around point Z. Point A is the reflection of point N across point Y. Let V be the midpoint of segment QA. Point F is the midpoint of segment YZ. Let S be the midpoint of segment VN. Let I be the reflection of point F across point A. What is the measure of angle NIS, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let ZT = 8. Let V be the reflection of point T across point Z. Point L is obtained by rotating point T by angle 7pi/12 about point V. Point N is the reflection of point T across point V. Let O be the rotation of point T by angle pi/6 around point L. Point I is the midpoint of segment VN. Let E be the orthocenter of triangle NVO. Let A be the reflection of point I across point E. Find the distance between points A and I. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
NF = 11. Let E be the rotation of point N by angle pi/4 around point F. Let R be the circumcircle of triangle EFN. Let Y be the incenter of triangle EFN. Let T be the inversion of point Y in circle R. Find the distance between points Y and T. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
RJ = 8. Point K is obtained by rotating point R by angle pi/12 about point J. Let O be the reflection of point J across point K. Let G be the rotation of point K by angle 11pi/12 around point O. Point T is the reflection of point O across point J. Circle Q is the incircle of triangle RGJ. Let I be the rotation of point O by angle 11pi/12 around point T. Let D be the inversion of point I in circle Q. Point E is the midpoint of segment RD. Let M be the inversion of point T in circle Q. What is the measure of angle MEK, 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 ZI = 11. Let K be the rotation of point Z by angle 7pi/12 around point I. Let S be the midpoint of segment ZK. Point F is the reflection of point I across segment SZ. Point U is the incenter of triangle ISZ. Find the length of segment UF. Express your answer as a decimal number rounded to 2 decimal places.

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IZ = 2. Point P is obtained by rotating point Z by angle 5pi/6 about point I. Point K is the reflection of point P across point Z. Point B is the centroid of triangle ZIP. Point O is the midpoint of segment IB. What is the inradius of triangle OKI? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let BQ = 2. Point T is the midpoint of segment QB. Let A be the vector from point T to point B. Let E be the midpoint of segment BT. Let Z be point B translated by vector A. Let L be the line through point Z perpendicular to segment QB. Let D be the line through point T perpendicular to segment QB. Point K lies on line L at distance 10 from point Z. Let J be the midpoint of segment BE. Line S is the reflection of line L across line D. Point O is the reflection of point B across line S. Point N is the reflection of point K across point O. Find the measure of angle KJN, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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ID = 5. Point R is obtained by rotating point D by angle 2pi/3 about point I. Point L is the midpoint of segment IR. Let H be the angle bisector of angle ILR. Point A lies on line H at distance 8 from point I. Point S is the midpoint of segment RA. Let E be the reflection of point S across point A. Let O be the centroid of triangle ISE. Find the distance between points O and L. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let OK = 4. Let Q be the midpoint of segment KO. Point F is obtained by rotating point O by angle 11pi/12 about point K. Let L be the reflection of point F across point O. Point X is obtained by rotating point F by angle pi/6 about point O. Let A be the reflection of point O across point X. Let R be the perpendicular bisector of segment LX. Let P be the line through point L perpendicular to line R. Let J be the perpendicular bisector of segment FO. Point B is the reflection of point Q across line J. Point E is the reflection of point A across line P. What is the circumradius of triangle EBX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
OW = 9. Point R is obtained by rotating point W by angle 5pi/6 about point O. Point D is the midpoint of segment OW. Let G be the midpoint of segment DW. Point L is the reflection of point G across segment RD. What is the circumradius of triangle LRW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let NH = 4. Let K be the rotation of point N by angle 2pi/3 around point H. Point M is the midpoint of segment KN. Let I be the centroid of triangle MHN. Let O be the incenter of triangle MHN. What is the distance between points I and O? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
FT = 3. Let N be the rotation of point T by angle 3pi/4 around point F. Let V be the rotation of point N by angle pi/6 around point T. Let Z be the angle bisector of angle VFT. Point J is the reflection of point F across point N. Point O is the reflection of point T across line Z. Point P is the incenter of triangle TJN. Find the inradius of triangle PNO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PF = 9. Point B is the reflection of point F across point P. Point Q is the reflection of point P across point B. Point U is obtained by rotating point F by angle pi/3 about point B. Line C is the angle bisector of angle BUQ. Let K be the midpoint of segment BU. Let R be the reflection of line C through point Q. Let H be the line through point K perpendicular to segment UQ. Line W is the reflection of line R across line H. Let X be the incenter of triangle UQP. Point M is the reflection of point P across line W. Point Y is the reflection of point M across line C. Let V be the reflection of point Q across point Y. What is the length of segment XV? Express your answer as a decimal number rounded to 2 decimal places.

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KB = 10. Point P is the midpoint of segment KB. Line H is the angle bisector of angle BKP. Let V be the reflection of point K across point B. Point Z is on line H with |ZP| = 9. Point G is obtained by rotating point Z by angle 7pi/12 about point P. Let F be the perpendicular bisector of segment PV. Point A is the midpoint of segment GZ. Point Q is the reflection of point K across line F. Let R be the midpoint of segment QA. What is the measure of angle ABR, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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GI = 5. Point Y is obtained by rotating point I by angle pi/3 about point G. Let R be the rotation of point Y by angle pi/4 around point I. Line B passes through point G perpendicular to segment IR. Let F be the midpoint of segment RY. Point C is the reflection of point F across line B. Find the measure of angle FCG, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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RY = 11. Point V is the midpoint of segment RY. Let X be the rotation of point R by angle pi/2 around point Y. Point O is the centroid of triangle VXY. Let G be the perpendicular bisector of segment VX. Line L is the reflection of line G across point R. Point K is the reflection of point O across line L. Point C is the midpoint of segment RX. Find the length of segment KC. Express your answer as a decimal number rounded to 2 decimal places.

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Let IF = 9. Let R be the rotation of point I by angle 5pi/6 around point F. Let H be the reflection of point R across point I. Let C be the circumcenter of triangle FIH. Let E be the midpoint of segment FI. Point U is the reflection of point C across point R. Find the distance between points E and U. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let LA = 7. Point K is the midpoint of segment LA. Point E is obtained by rotating point K by angle pi/2 about point L. Point I is the midpoint of segment KL. Point D is the centroid of triangle LIE. Let T be the midpoint of segment EL. Let B be the reflection of point T across point D. Point C is the midpoint of segment IE. What is the length of segment BC? Express your answer as a decimal number rounded to 2 decimal places.

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GV = 9. Point D is obtained by rotating point G by angle 7pi/12 about point V. Point E is the reflection of point V across point D. Point N is obtained by rotating point D by angle 11pi/12 about point E. Point K is the orthocenter of triangle NED. Circle S is the circumcircle of triangle NED. Let P be the angle bisector of angle DKV. Let B be the inversion of point V in circle S. Point T is the reflection of point V across line P. Let O be the reflection of point B across point E. Find the length of segment TO. Express your answer as a decimal number rounded to 2 decimal places.

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Let UY = 7. Let N be the midpoint of segment YU. Point P is obtained by rotating point N by angle 7pi/12 about point U. Circle M is the incircle of triangle NUP. Let J be the reflection of circle M through point Y. Let V be the rotation of point Y by angle 5pi/6 around point U. Let C be the midpoint of segment PU. Let L be the circumcenter of triangle NUP. Point D is the inversion of point V with respect to circle J. Let W be the midpoint of segment PN. Let R be the line through point U parallel to segment CD. Point O is the reflection of point L across line R. What is the length of segment OW? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
OC = 9. Point P is obtained by rotating point O by angle 3pi/4 about point C. Point K is obtained by rotating point P by angle 11pi/12 about point O. Let W be the midpoint of segment CK. Let F be the centroid of triangle WOP. Point L is the reflection of point F across segment OP. Find the measure of angle FLW, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
MG = 10. Point P is the reflection of point G across point M. Let K be the rotation of point M by angle pi/6 around point G. Point I is the reflection of point G across segment PK. Line V passes through point P perpendicular to segment PK. Let O be the line through point I parallel to line V. Point C is the midpoint of segment PK. Point E is the midpoint of segment CG. Let F be the reflection of point E across point K. Point A is the reflection of point M across line O. What is the measure of angle FPA, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
KF = 3. Line E is the perpendicular bisector of segment KF. Point A lies on line E at distance 6 from point K. Point L is the incenter of triangle KFA. Point B is the orthocenter of triangle KFA. Line I passes through point L parallel to segment KF. Segment FA and line I intersect at point W. Find the distance between points W and B. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
YK = 6. Point U is obtained by rotating point K by angle 2pi/3 about point Y. Point A is obtained by rotating point K by angle 11pi/12 about point U. Point J is the reflection of point A across point U. Let Q be the centroid of triangle KYJ. Let Z be the line through point U perpendicular to segment JK. Point W is the reflection of point Q across line Z. Point F is the midpoint of segment YW. Find the inradius of triangle FAW. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let FA = 6. Let U be the rotation of point A by angle pi/12 around point F. Let P be the rotation of point F by angle 2pi/3 around point A. Let G be the reflection of point A across point P. Point K is the midpoint of segment PG. Let Q be the midpoint of segment KF. Point Z is the reflection of point G across segment KF. Find the measure of angle ZUQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ZC = 4. Let J be the midpoint of segment ZC. Point L is obtained by rotating point Z by angle 7pi/12 about point J. Point R is obtained by rotating point J by angle 3pi/4 about point C. Let Y be the perpendicular bisector of segment JL. Point Q is the reflection of point Z across line Y. Point P is the circumcenter of triangle JLR. Find the inradius of triangle PQZ. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let US = 10. Point V is obtained by rotating point U by angle pi/12 about point S. Point O is the reflection of point U across point S. Let B be the midpoint of segment SO. Let N be the centroid of triangle OVU. Point J is the reflection of point N across segment VU. Find the distance between points J and B. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let GT = 12. Let A be the rotation of point T by angle pi/4 around point G. Point O is the orthocenter of triangle GTA. Let V be the incenter of triangle GTA. Let F be the midpoint of segment TA. Point C is the midpoint of segment VF. Point R is the midpoint of segment CT. Point L is on segment TA. Let P be the midpoint of segment RO. Find the distance between points C and P. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
NZ = 3. Let L be the midpoint of segment NZ. Line E is the perpendicular bisector of segment ZN. Line X passes through point N parallel to line E. Let M be the reflection of line X across line E. Point D lies on line M at distance 5 from point N. Point P lies on line X. Let T be the reflection of point L across point P. Point V is the centroid of triangle DNT. Find the measure of angle VTL, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let IM = 11. Let H be the midpoint of segment IM. Point P is obtained by rotating point H by angle 11pi/12 about point I. Let O be the reflection of point P across point M. Point V is the orthocenter of triangle IOP. Point R is the reflection of point O across segment VP. What is the distance between points P and R? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
YG = 4. Let N be the vector from point G to point Y. Point U is the translation of point Y by vector N. Let T be the reflection of point G across point U. Let V be the rotation of point T by angle 5pi/12 around point Y. Let B be the rotation of point U by angle pi/12 around point V. Line X is the angle bisector of angle TBG. Let L be the line through point Y parallel to line X. Line W passes through point T parallel to line L. Point F is the reflection of point B across line W. Find the measure of angle VTF, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let GZ = 10. Point X is obtained by rotating point Z by angle pi/12 about point G. Point Q is the midpoint of segment GX. Let M be the reflection of point Q across point Z. Circle R is the circumcircle of triangle GXZ. Point C is the incenter of triangle GXZ. Point D is the reflection of point M across segment XZ. Point B is the inversion of point Q with respect to circle R. Find the inradius of triangle DBC. Express your answer as a decimal number rounded to 2 decimal places.

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