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WG = 10. Point Y is obtained by rotating point G by angle 7pi/12 about point W. Let I be the rotation of point W by angle 3pi/4 around point G. Point A is the reflection of point G across point I. Point O is the reflection of point A across segment GW. Let H be the midpoint of segment YI. Let X be the circumcenter of triangle OYI. Point Q is the midpoint of segment XY. What is the measure of angle QHA, 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 AF = 12. Let L be the rotation of point A by angle pi/4 around point F. Let J be the line through points L and F. Point C is the midpoint of segment AF. Line H is the reflection of line J across point A. Point R is the reflection of point L across line H. Point Y is the circumcenter of triangle LAC. Let U be the circumcircle of triangle LAC. Let W be the inversion of point Y in circle U. What is the measure of angle WRL, 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 PR = 8. Let C be the rotation of point R by angle 5pi/6 around point P. Point W is the midpoint of segment CR. Point T is the midpoint of segment PC. Point M is the reflection of point T across segment PW. What is the length of segment MP? 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. Point B is the reflection of point U across point D. Let Z be the rotation of point D by angle pi/12 around point B. Point S is the centroid of triangle ZUB. Let X be the incenter of triangle ZUB. Point W is the midpoint of segment XU. Find the measure of angle WSX, 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 HE = 9. Let L be the rotation of point H by angle pi/4 around point E. Let V be the reflection of point E across point L. Point D is obtained by rotating point L by angle pi about point H. Let G be the midpoint of segment DV. Let P be the incenter of triangle LED. Let C be the reflection of point G across point H. Find the measure of angle HCP, 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 AJ = 4. Line Z is the perpendicular bisector of segment JA. Point W lies on line Z at distance 8 from point J. Point F is the midpoint of segment AW. Point E is the reflection of point W across point J. Point S is the orthocenter of triangle FWE. What is the distance between points A and S? Express your answer as a decimal number rounded to 2 decimal places.

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


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Let GP = 5. Line L is the perpendicular bisector of segment GP. Let A be the line through points G and P. Let H be the line through point G perpendicular to line A. Point T is the reflection of point P across point G. Let O be the rotation of point G by angle 5pi/6 around point P. Line E is the reflection of line H across line L. Let F be the rotation of point P by angle 3pi/4 around point G. Point C is the reflection of point G across line E. Point V is the reflection of point O across segment GF. Let N be the midpoint of segment CV. What is the distance between points N and T? 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 FR = 4. Let P be the vector from point R to point F. Point H is obtained by rotating point R by angle pi/4 about point F. Let Q be the angle bisector of angle HRF. Let Y be point H translated by vector P. Let K be the line through points H and Y. Line D is the reflection of line Q across point F. Let S be point F translated by vector P. Point Z is the reflection of point R across line K. Let E be the line through point Z parallel to line D. Point G is the midpoint of segment SF. Circle L is the circumcircle of triangle SFH. Let T be the reflection of circle L across line E. Point B is the inversion of point G with respect to circle T. What is the measure of angle YBF, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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OE = 11. L is placed 3 units away from E. Point Q is obtained by rotating point E by angle pi/3 about point L. Point U is obtained by rotating point L by angle 2pi/3 about point Q. Line G is the perpendicular bisector of segment EL. Point Z is the reflection of point U across line G. Let H be the reflection of point Z across point E. Let J be the midpoint of segment ZQ. Point F is the reflection of point J across segment EL. What is the distance between points F and H? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let X be the orthocenter of triangle B. Point F is the reflection of point Y across point B. Point D is the midpoint of segment XF. Point N is the reflection of point B across point D. What is the distance between points Y and N? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let LU = 2. Let J be the midpoint of segment UL. Point G is obtained by rotating point J by angle 11pi/12 about point L. Let H be the perpendicular bisector of segment GU. Line N is the reflection of line H across point G. Point Z is the reflection of point G across line N. Point M is the reflection of point Z across line H. Point V is the reflection of point L across point M. Let A be the midpoint of segment GL. Find the measure of angle UVA, 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 LC = 5. Point Q is the midpoint of segment LC. Let D be the rotation of point C by angle pi/3 around point Q. Let S be the reflection of point Q across point D. Let W be the circumcenter of triangle DCS. Let Y be the circumcircle of triangle DCS. Let T be the reflection of circle Y through point L. Let F be the inversion of point S in circle T. Find the measure of angle WFQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
KI = 11. Point A is obtained by rotating point I by angle 11pi/12 about point K. Let V be the reflection of point I across point A. Point F is the reflection of point A across point I. Point X is the orthocenter of triangle AKI. Let O be the perpendicular bisector of segment AX. Point D is the reflection of point V across line O. What is the distance between points D and F? Express your answer as a decimal number rounded to 2 decimal places.

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Let HI = 2. Let V be the reflection of point I across point H. Let E be the reflection of point H across point I. Point J is obtained by rotating point V by angle 5pi/12 about point H. Point B is obtained by rotating point J by angle pi/3 about point V. Point C is the reflection of point J across point I. Let S be the incenter of triangle BCE. Circle M is the circumcircle of triangle BCE. Let Y be the inversion of point S in circle M. What is the length of segment YC? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WZ = 3. Point Q is obtained by rotating point Z by angle pi/2 about point W. Point L is the incenter of triangle QWZ. Circle E is the circumcircle of triangle QWZ. Let N be the reflection of point L across point Q. Let H be the inversion of point N in circle E. Find the length of segment NH. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BP = 11. Let D be the rotation of point P by angle 11pi/12 around point B. Let A be the midpoint of segment PD. Circle G is the incircle of triangle DAB. Let C be the circumcenter of triangle DAB. Point E is the inversion of point P with respect to circle G. Line M passes through points E and A. Point F is the intersection of line M and segment BD. Let L be the midpoint of segment CA. What is the length of segment FL? 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 RK = 10. Let S be the rotation of point R by angle 7pi/12 around point K. Let J be the incenter of triangle SKR. Let A be the centroid of triangle SKR. Point H is the midpoint of segment JR. Find the measure of angle RAH, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let EO = 5. Let L be the rotation of point O by angle pi/3 around point E. Let F be the reflection of point O across point L. Point Q is obtained by rotating point E by angle 7pi/12 about point O. Line A is the perpendicular bisector of segment OF. Point I is the reflection of point Q across line A. Point Z is the midpoint of segment IQ. What is the inradius of triangle ZFQ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QN = 4. Point M is the midpoint of segment QN. Point G is obtained by rotating point Q by angle 2pi/3 about point M. Let H be the rotation of point N by angle 3pi/4 around point M. Point Z is the midpoint of segment MQ. Point J is the reflection of point G across segment MH. What is the inradius of triangle ZMJ? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let VO = 1. Line U passes through points O and V. Let X be the rotation of point V by angle pi/6 around point O. Line H passes through point X perpendicular to line U. Let F be the rotation of point X by angle 2pi/3 around point V. Let J be the line through point F perpendicular to line H. Point W is the reflection of point X across line J. Let M be the orthocenter of triangle OFW. What is the measure of angle FXM, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
XV = 10. Point W is obtained by rotating point X by angle pi/6 about point V. Point R is the midpoint of segment WX. Let C be the orthocenter of triangle WXV. Point F is the reflection of point R across segment XV. Find the distance between points F and C. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
YE = 2. Line N is the perpendicular bisector of segment EY. Let P be the reflection of point E across point Y. Point S lies on line N at distance 1 from point E. Point D is the midpoint of segment ES. Let L be the reflection of point E across point P. Point R is the midpoint of segment LD. Let J be the midpoint of segment DR. Find the measure of angle RJP, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let BV = 7. Let S be the rotation of point V by angle 5pi/12 around point B. Let M be the rotation of point B by angle 2pi/3 around point V. Point C is the centroid of triangle MBS. Point W is the orthocenter of triangle MBS. Point Q is the incenter of triangle MBS. Line H is the perpendicular bisector of segment CQ. Point J is the reflection of point M across line H. Find the distance between points W and J. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
PB = 11. Line D is the perpendicular bisector of segment BP. Let Y be the midpoint of segment PB. Point V lies on line D at distance 5 from point Y. Let C be the reflection of line D through point B. Point U is the reflection of point P across line C. Let T be the midpoint of segment PU. Let J be the centroid of triangle PUV. Find the length of segment JT. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let QY = 7. Point E is the reflection of point Q across point Y. Point R is the reflection of point E across point Q. Line I is the perpendicular bisector of segment RQ. Point Z lies on line I at distance 9 from point Q. Let T be the reflection of point Z across point E. Point A is the midpoint of segment YR. Let S be the midpoint of segment QT. Point F is the midpoint of segment QA. Find the measure of angle ZFS, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FI = 10. Let V be the reflection of point I across point F. Let Y be the reflection of point F across point V. Let W be the rotation of point Y by angle 3pi/4 around point V. Point T is the reflection of point F across point Y. Let X be the midpoint of segment TW. Let S be the reflection of point F across point T. What is the inradius of triangle WSX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
VR = 10. Let W be the line through point R perpendicular to segment RV. Point Z is obtained by rotating point R by angle pi/4 about point V. Let P be the midpoint of segment ZR. Line A passes through point P parallel to line W. Point D is the midpoint of segment RV. Point H is the midpoint of segment ZD. Point K is on segment RV. Point U is the reflection of point K across line A. What is the measure of angle UHP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LS = 9. Let W be the rotation of point L by angle pi/2 around point S. Let M be the reflection of point W across point L. Let T be the centroid of triangle WLS. Point A is the reflection of point T across segment LS. What is the distance between points A and M? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
GP = 8. Point D is obtained by rotating point P by angle pi/12 about point G. Line J passes through points D and G. Point C is the reflection of point D across segment GP. Point W is the reflection of point P across line J. Let N be the midpoint of segment DP. What is the inradius of triangle CNG? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Point H is the midpoint of segment TI24. Let N be the midpoint of segment GH. Point I is the reflection of point G across point L. Point R is the midpoint of segment HI. Find the distance between points N and R. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HW = 5. Let R be the rotation of point W by angle pi/3 around point H. Let X be the rotation of point R by angle 11pi/12 around point H. Point B is the midpoint of segment WX. Let M be the midpoint of segment XR. Point G is the reflection of point B across segment HX. Let P be the incenter of triangle HXM. Let L be the midpoint of segment GR. What is the measure of angle XPL, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
LS = 4. Let G be the midpoint of segment LS. Line X passes through point G perpendicular to segment LS. Let Z be the reflection of point S across point L. Let O be the reflection of point G across point S. Let H be the line through point S parallel to line X. Line T is the reflection of line X across line H. Point A is obtained by rotating point S by angle 5pi/12 about point O. Let B be the angle bisector of angle OAZ. Line Y is the reflection of line T across point G. Let J be the rotation of point G by angle pi/4 around point L. Point D is the reflection of point A across segment LS. Point K is the reflection of point L across line B. Let E be the line through point J perpendicular to line Y. Line W is the angle bisector of angle DSO. Point Q is the reflection of point K across line E. Point F is the reflection of point D across line W. Let I be the reflection of point F across point K. Find the measure of angle FIQ, 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 OZ = 11. Line S is the perpendicular bisector of segment OZ. Point W is on line S with |WO| = 11. Point K is the circumcenter of triangle ZOW. Point J is the incenter of triangle ZOW. Point P is the reflection of point J across segment WZ. Find the distance between points K and P. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let NI = 2. Let A be the rotation of point I by angle pi/2 around point N. Point H is the midpoint of segment NA. Point K is the reflection of point H across point I. Let F be the perpendicular bisector of segment KH. Point R is the reflection of point N across line F. Point X is the centroid of triangle IHR. What is the length of segment NX? 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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CT = 9. Let V be the reflection of point C across point T. Let K be the midpoint of segment VT. Let S be the rotation of point V by angle 5pi/12 around point T. Point G is the orthocenter of triangle SKV. Let H be the midpoint of segment KV. Point L is the reflection of point G across point C. Let M be the reflection of point T across point L. Find the distance between points H and M. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let SD = 8. Line Z is the perpendicular bisector of segment DS. Point E is on line Z with |ES| = 3. Line B passes through point S parallel to segment DE. Point C is the reflection of point D across line B. Point Q is the midpoint of segment ES. Find the length of segment CQ. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let DP = 11. Point K is obtained by rotating point P by angle pi/6 about point D. Point Y is the reflection of point D across point K. Circle F is the incircle of triangle PKY. Let N be the inversion of point P in circle F. What is the distance between points D and N? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let HV = 3. Point R is obtained by rotating point V by angle pi/2 about point H. Point Z is obtained by rotating point V by angle pi/6 about point R. Let U be the midpoint of segment VZ. Point I is the reflection of point H across segment ZR. Let L be the midpoint of segment IZ. Find the measure of angle RLU, 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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IO = 9. Point P is obtained by rotating point O by angle 2pi/3 about point I. Point X is the circumcenter of triangle PIO. Point U is the incenter of triangle PIO. Point V is the reflection of point P across point U. Circle D is the circumcircle of triangle PIO. Let C be the inversion of point V in circle D. Find the length of segment XC. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Point Y is the midpoint of segment IA. Let L be the angle bisector of angle UIY. Point M is the circumcenter of triangle I. Point W is the midpoint of segment MU. Point O is the reflection of point W across point Y. Point X is the reflection of point O across line L. What is the length of segment AX? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
EX = 4. Let B be the midpoint of segment EX. Line S is the perpendicular bisector of segment BX. Line M is the reflection of line S across point X. Point N is on line M with |NE| = 7. Point R is the reflection of point X across point N. Point D is the reflection of point E across segment NB. Find the inradius of triangle XRD. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
FQ = 6. Point K is the midpoint of segment QF. Let U be the rotation of point K by angle pi/12 around point F. Point Z is obtained by rotating point Q by angle 3pi/4 about point F. Let A be the reflection of point Q across point Z. Point B is the reflection of point A across segment UZ. Find the circumradius of triangle FAB. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let RV = 1. Let N be the perpendicular bisector of segment RV. Point C lies on line N at distance 10 from point V. Point P is the reflection of point C across point R. Let M be the midpoint of segment RP. Point T is the midpoint of segment PV. Let I be the incenter of triangle MPV. Point K is the reflection of point T across segment RV. Point X is the midpoint of segment KR. What is the length of segment IX? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let TR = 4. Let W be the rotation of point T by angle pi around point R. Let X be the reflection of point R across point T. Let U be the midpoint of segment RW. Let V be the vector from point U to point R. Point M is obtained by rotating point X by angle pi/12 about point U. Let E be point X translated by vector V. Let J be point E translated by vector V. Point A is obtained by rotating point J by angle pi/12 about point E. Circle N is the incircle of triangle AEM. Point D is the midpoint of segment EM. Let Z be the inversion of point D in circle N. Find the length of segment ZA. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
CY = 6. Point S is obtained by rotating point C by angle 11pi/12 about point Y. Point H is obtained by rotating point C by angle 2pi/3 about point S. Point F is the orthocenter of triangle SCH. Let M be the line through point Y perpendicular to segment SC. Line Z passes through point S parallel to line M. Point W is the reflection of point F across line Z. Find the distance between points W and F. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
QS = 3. Let H be the rotation of point S by angle 3pi/4 around point Q. Let V be the reflection of point S across point H. Let A be the incenter of triangle HVQ. Let K be the midpoint of segment QH. Find the distance between points K and A. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
WD = 12. Let P be the rotation of point D by angle 2pi/3 around point W. Line H is the angle bisector of angle DPW. Point T is the reflection of point W across line H. Let Y be the midpoint of segment DW. What is the inradius of triangle PTY? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
AX = 3. Point J is the reflection of point A across point X. Line N is the angle bisector of angle AJX. Let U be the rotation of point J by angle pi/4 around point X. Point H is the reflection of point U across line N. Point O is the midpoint of segment HU. Point I is the reflection of point J across point O. Point G is the midpoint of segment HJ. Find the measure of angle IUG, 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 IT = 11. Point C is the midpoint of segment TI. Let B be the rotation of point C by angle pi/12 around point I. Let H be the rotation of point T by angle 11pi/12 around point B. Let P be the rotation of point B by angle 11pi/12 around point C. Let W be the angle bisector of angle CPH. Point L is the reflection of point T across line W. Point F is the reflection of point H across point T. Line Z is the angle bisector of angle IFB. Point Q is the reflection of point H across line Z. Point R is the reflection of point Q across line W. What is the measure of angle LRC, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
ZF = 4. Let X be the reflection of point F across point Z. Point O is obtained by rotating point Z by angle pi/3 about point F. Point E is the centroid of triangle OXZ. Let C be the perpendicular bisector of segment EO. Point K is the reflection of point Z across line C. What is the distance between points K and O? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Point R is the orthocenter of triangle J. Let Y be the circumcircle of triangle J. Circle T is the reflection of circle Y across point P. Let G be the inversion of point P in circle T. Find the length of segment RG. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let TY = 1. Point U is the midpoint of segment YT. Let H be the rotation of point U by angle pi/12 around point Y. Point B is obtained by rotating point Y by angle pi/6 about point H. Let G be the rotation of point T by angle pi/2 around point U. Point L is the centroid of triangle TYB. Let J be the midpoint of segment BU. Let O be the orthocenter of triangle LUG. Find the circumradius of triangle JGO. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MV = 8. Let S be the rotation of point M by angle pi/6 around point V. Point J is obtained by rotating point V by angle pi about point M. Line N is the perpendicular bisector of segment VS. Point U is the reflection of point M across line N. Let P be the centroid of triangle MSU. What is the distance between points P and J? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
UR = 6. Point E is obtained by rotating point U by angle 5pi/12 about point R. Let G be the rotation of point U by angle pi/4 around point R. Point F is the reflection of point G across point E. Let W be the reflection of point U across point E. Let M be the orthocenter of triangle GFR. What is the length of segment MW? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let DZ = 8. Let K be the rotation of point Z by angle 3pi/4 around point D. Let P be the reflection of point Z across point K. Let I be the reflection of point K across point D. Line A passes through point Z parallel to segment KI. Line B passes through point P parallel to segment IZ. Let O be the reflection of line B across line A. Line W is the reflection of line A across line O. Let N be the midpoint of segment IZ. Point T is the reflection of point N across line W. What is the distance between points P and T? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
BH = 2. Let F be the rotation of point B by angle 11pi/12 around point H. Let Y be the rotation of point H by angle 11pi/12 around point B. Point G is the reflection of point Y across point B. Point E is the midpoint of segment HF. Point U is the reflection of point G across segment HF. Point I is the orthocenter of triangle EUB. Let X be the centroid of triangle EUB. Find the distance between points X and I. Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
MF = 6. Let K be the reflection of point F across point M. Line G passes through point K perpendicular to segment FM. Point Q lies on line G. S is constructed so that QS = 9. Point O lies on line G. Point E is the reflection of point O across segment FM. Let V be the midpoint of segment SE. Let T be the midpoint of segment EQ. What is the length of segment TV? Express your answer as a decimal number rounded to 2 decimal places.

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NX = 1. Let D be the rotation of point X by angle 3pi/4 around point N. Point P is obtained by rotating point X by angle pi/4 about point N. Let I be the midpoint of segment XD. Point M is the circumcenter of triangle NIP. Point B is the orthocenter of triangle NIP. Find the length of segment MB. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let PM = 9. Point H is obtained by rotating point M by angle 2pi/3 about point P. Point W is the reflection of point H across point M. Point E is the incenter of triangle HMP. Point X is the reflection of point W across segment PH. What is the distance between points E and X? Express your answer as a decimal number rounded to 2 decimal places.

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


[User]
Let MA = 11. Let R be the vector from point A to point M. Point U is obtained by rotating point A by angle 7pi/12 about point M. Point D is the translation of point M by vector R. Let K be the line through point U perpendicular to segment UM. Let Z be the circumcenter of triangle AUM. Let Y be the line through point Z parallel to line K. Point S is the reflection of point D across line Y. What is the measure of angle MDS, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
XW = 3. Point J is the reflection of point X across point W. Let H be the midpoint of segment WX. Let S be the midpoint of segment HJ. Point F is obtained by rotating point X by angle pi/2 about point W. Let Q be the angle bisector of angle HFS. Let G be the reflection of point F across point X. Line N passes through point X parallel to line Q. Line N and segment HG intersect at point B. What is the distance between points B and J? Express your answer as a decimal number rounded to 2 decimal places.

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Let VQ = 4. Let J be the perpendicular bisector of segment QV. Point W lies on line J. Let S be the midpoint of segment QW. Let X be the line through point Q parallel to line J. Point L is the reflection of point V across line X. Point D is the reflection of point V across point S. Point Z is the reflection of point W across segment LD. Find the distance between points Z and D. Express your answer as a decimal number rounded to 2 decimal places.

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Let HD = 7. Point V is obtained by rotating point D by angle pi/6 about point H. Point E is obtained by rotating point D by angle 5pi/12 about point V. Let I be the reflection of point H across point V. Let M be the circumcenter of triangle HDE. Point S is the midpoint of segment HD. What is the circumradius of triangle IMS? Express your answer as a decimal number rounded to 2 decimal places.

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Let RW = 11. Point E is the midpoint of segment RW. Let Q be the rotation of point E by angle 2pi/3 around point R. Point L is the orthocenter of triangle WEQ. Let Y be the reflection of point R across point L. Let O be the midpoint of segment LW. Let P be the centroid of triangle WYL. What is the distance between points P and O? Express your answer as a decimal number rounded to 2 decimal places.

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IG = 1. Let N be the reflection of point G across point I. Let R be the line through points G and I. Let Q be the rotation of point N by angle pi around point G. Let J be the reflection of point I across point G. Point C is obtained by rotating point Q by angle 5pi/12 about point J. Line H is the reflection of line R across point C. Let S be the reflection of line H through point N. Let L be the midpoint of segment IN. Point V is the reflection of point L across line H. Point D is the reflection of point V across line S. Point Z is the midpoint of segment CG. Find the measure of angle JDZ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let IL = 4. Point R is the reflection of point L across point I. Let G be the rotation of point R by angle pi/2 around point L. Point P is the centroid of triangle IGL. Point V is the incenter of triangle IGL. Point J is the reflection of point V across point I. What is the measure of angle JVP, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let GB = 6. Point F is the midpoint of segment BG. Line Z is the perpendicular bisector of segment FB. Point U is the reflection of point F across point G. Point A is obtained by rotating point F by angle pi/12 about point U. Point H is the reflection of point U across line Z. Point Q is the reflection of point A across point H. Let S be the circumcenter of triangle GQA. Find the length of segment AS. Express your answer as a decimal number rounded to 2 decimal places.

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BT = 1. Let D be the reflection of point T across point B. Point K is the midpoint of segment BD. Point X is obtained by rotating point K by angle 2pi/3 about point B. Let Q be the reflection of point X across point T. Point O is the orthocenter of triangle QXD. Let W be the midpoint of segment DQ. Point P is the circumcenter of triangle QXD. Find the measure of angle OPW, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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HD = 7. Let P be the line through points H and D. Let K be the rotation of point H by angle pi/4 around point D. Line Y passes through point K perpendicular to line P. Point F is obtained by rotating point H by angle 5pi/12 about point K. Point V is the reflection of point D across line Y. Point W is the midpoint of segment KV. Let U be the midpoint of segment WF. Point B is the reflection of point V across segment UF. What is the measure of angle BFK, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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VN = 5. Point C is the midpoint of segment VN. Let R be the rotation of point C by angle pi/4 around point V. Point O is the reflection of point N across segment RC. Point U is the reflection of point R across point C. Find the inradius of triangle OUN. Express your answer as a decimal number rounded to 2 decimal places.

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Let IB = 5. Point U is obtained by rotating point I by angle 5pi/6 about point B. Let V be the centroid of triangle UIB. Point S is the midpoint of segment IB. Let M be the incircle of triangle UIB. Point E is the inversion of point S with respect to circle M. What is the distance between points E and V? Express your answer as a decimal number rounded to 2 decimal places.

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Let OQ = 5. Point K is the midpoint of segment OQ. Let R be the rotation of point K by angle 5pi/6 around point O. Let W be the rotation of point O by angle 5pi/12 around point R. Let Y be the incircle of triangle WOR. Let E be the midpoint of segment OR. Point J is the reflection of point K across segment OR. Let I be the inversion of point J in circle Y. Let G be the circumcenter of triangle WOR. Find the measure of angle EIG, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let QF = 5. Z is placed 4 units away from F. Point P is obtained by rotating point F by angle 11pi/12 about point Q. Let U be the midpoint of segment PF. Let O be the midpoint of segment QZ. Point Y is the midpoint of segment ZU. What is the length of segment OY? Express your answer as a decimal number rounded to 2 decimal places.

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BV = 1. Point E is the midpoint of segment BV. Let W be the rotation of point E by angle pi/2 around point B. Point K is obtained by rotating point V by angle 5pi/12 about point E. Circle G is the incircle of triangle KBW. Point F is the inversion of point B with respect to circle G. Let T be the incenter of triangle KBW. Let N be the circumcenter of triangle FKE. Find the length of segment TN. Express your answer as a decimal number rounded to 2 decimal places.

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JN = 12. Let Y be the rotation of point N by angle 7pi/12 around point J. Let Z be the reflection of point Y across point J. Let G be the centroid of triangle YJN. Let H be the midpoint of segment NG. Point B is the midpoint of segment JG. What is the inradius of triangle ZHB? Express your answer as a decimal number rounded to 2 decimal places.

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Let RH = 9. Point W is the midpoint of segment RH. Point B is obtained by rotating point W by angle pi/4 about point R. Line X is the angle bisector of angle WBR. Let G be the reflection of line X through point R. Point J is the reflection of point W across line G. Point U is the reflection of point R across line G. Find the circumradius of triangle UHJ. Express your answer as a decimal number rounded to 2 decimal places.

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MP = 12. Let H be the line through points P and M. Let T be the rotation of point M by angle pi/3 around point P. Point L is the reflection of point T across line H. Let V be the reflection of point M across point L. Let Z be the orthocenter of triangle MTV. What is the length of segment VZ? Express your answer as a decimal number rounded to 2 decimal places.

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RK = 9. Let A be the rotation of point R by angle pi/4 around point K. Let W be the line through points A and R. Point M is obtained by rotating point K by angle 11pi/12 about point R. Point P is the reflection of point M across line W. Point X is the reflection of point A across segment RK. Let J be the orthocenter of triangle MPK. What is the inradius of triangle PXJ? Express your answer as a decimal number rounded to 2 decimal places.

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Let WZ = 4. Let O be the reflection of point Z across point W. Let K be the rotation of point Z by angle 7pi/12 around point W. Let X be the perpendicular bisector of segment OW. Let D be the circumcenter of triangle KWO. Point A is the reflection of point D across line X. Let E be the midpoint of segment AO. What is the measure of angle ZEA, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let EB = 12. Line U is the perpendicular bisector of segment EB. Point T is obtained by rotating point B by angle 11pi/12 about point E. Let A be the line through point B parallel to line U. Line P is the reflection of line A across line U. Point D is the reflection of point B across line P. Point Z is the circumcenter of triangle ETB. What is the distance between points D and Z? Express your answer as a decimal number rounded to 2 decimal places.

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Let EH = 7. Point M is obtained by rotating point E by angle 2pi/3 about point H. Point J is the reflection of point H across point E. Let O be the perpendicular bisector of segment HM. Point B is the reflection of point J across line O. Let W be the reflection of point J across point B. Find the measure of angle MWH, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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IY = 7. Point H is obtained by rotating point I by angle 5pi/12 about point Y. Point C is the midpoint of segment IH. Let U be the midpoint of segment YI. Point L is the centroid of triangle HYI. Let N be the midpoint of segment LU. Point Z is the reflection of point Y across point N. Find the distance between points Z and C. Express your answer as a decimal number rounded to 2 decimal places.

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QL = 3. Let S be the rotation of point Q by angle pi/3 around point L. I is constructed so that LI = 12. Point K is the midpoint of segment IQ. Point U is the midpoint of segment KL. Let F be the circumcenter of triangle KUS. Find the length of segment FU. Express your answer as a decimal number rounded to 2 decimal places.

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Let FX = 9. Let T be the rotation of point X by angle pi/2 around point F. Let C be the circumcenter of triangle XFT. Let Y be the midpoint of segment XC. Point J is the reflection of point T across point X. What is the distance between points Y and J? Express your answer as a decimal number rounded to 2 decimal places.

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AT = 8. Let S be the perpendicular bisector of segment AT. Point K lies on line S at distance 12 from point A. Point Y is the incenter of triangle KAT. Let W be the circumcenter of triangle KAT. Point F is the midpoint of segment AY. What is the distance between points W and F? Express your answer as a decimal number rounded to 2 decimal places.

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Let TE = 10. Let Y be the midpoint of segment TE. Point S is obtained by rotating point Y by angle 11pi/12 about point T. Let U be the midpoint of segment SE. Let N be the reflection of point T across point S. Point J is the reflection of point U across point Y. Point F is the reflection of point J across segment YS. What is the length of segment NF? Express your answer as a decimal number rounded to 2 decimal places.

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Let UC = 2. Point L is obtained by rotating point U by angle 2pi/3 about point C. Point P is the centroid of triangle LUC. Let N be the circumcircle of triangle LUC. Let I be the inversion of point P in circle N. What is the distance between points I and U? Express your answer as a decimal number rounded to 2 decimal places.

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JL = 12. Point O is obtained by rotating point J by angle 2pi/3 about point L. Let D be the reflection of point O across point J. Let N be the circumcenter of triangle OJL. Find the inradius of triangle DLN. Express your answer as a decimal number rounded to 2 decimal places.

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Let RU = 6. Let H be the midpoint of segment UR. Point C is obtained by rotating point R by angle pi/2 about point H. Point T is the reflection of point C across segment UR. Point K is the reflection of point R across point U. Let Y be the reflection of point T across point K. Let O be the midpoint of segment YR. What is the inradius of triangle YTO? Express your answer as a decimal number rounded to 2 decimal places.

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DY = 9. Point L is the midpoint of segment DY. Let G be the rotation of point L by angle 7pi/12 around point Y. Point C is the reflection of point G across point D. Point V is the reflection of point C across segment LD. Point M is the midpoint of segment CD. Find the inradius of triangle MLV. Express your answer as a decimal number rounded to 2 decimal places.

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Let LU = 6. Let X be the rotation of point L by angle pi/12 around point U. Let Y be the rotation of point L by angle pi/6 around point U. Let K be the midpoint of segment YL. Point C is the midpoint of segment XY. Let R be the midpoint of segment LX. What is the measure of angle CRK, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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XS = 10. Point E is the midpoint of segment XS. Let I be the rotation of point S by angle pi/12 around point E. Let Y be the perpendicular bisector of segment SX. Point B is the reflection of point I across line Y. Let Q be the circumcenter of triangle EXI. Let V be the orthocenter of triangle XBQ. What is the length of segment VX? Express your answer as a decimal number rounded to 2 decimal places.

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Let CO = 4. Let A be the rotation of point O by angle 7pi/12 around point C. Let Q be the incenter of triangle OAC. Let N be the reflection of point Q across point O. Point X is the orthocenter of triangle OAC. What is the length of segment NX? Express your answer as a decimal number rounded to 2 decimal places.

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Let HT = 8. Let M be the rotation of point H by angle 11pi/12 around point T. Point J is the midpoint of segment TM. Let I be the line through points H and J. Let B be the reflection of line I through point T. Line D passes through point J perpendicular to line I. Line P passes through point T parallel to line D. Point Z is the intersection of lines P and B. Find the measure of angle HMZ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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OB = 5. Line L is the perpendicular bisector of segment OB. Point N is obtained by rotating point B by angle 11pi/12 about point O. Let A be the rotation of point N by angle 7pi/12 around point B. Point T is obtained by rotating point A by angle pi/4 about point N. Point S is the reflection of point O across point N. Let Q be the rotation of point O by angle 11pi/12 around point N. Line J is the reflection of line L across point A. Point H is the reflection of point S across point T. Line J and segment OH intersect at point G. Find the length of segment GS. Express your answer as a decimal number rounded to 2 decimal places.

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DY = 2. Let Q be the rotation of point D by angle pi/6 around point Y. Point N is the midpoint of segment YQ. Point K is the midpoint of segment NY. Point A is the reflection of point Y across point D. Let W be the midpoint of segment KD. What is the measure of angle WDA, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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XD = 2. Point O is obtained by rotating point X by angle 11pi/12 about point D. Let V be the midpoint of segment DO. Point T is the reflection of point V across segment XD. Let Q be the reflection of point T across point D. What is the inradius of triangle DQV? Express your answer as a decimal number rounded to 2 decimal places.

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JQ = 9. Let D be the rotation of point Q by angle pi/12 around point J. Let S be the angle bisector of angle JQD. Point K is the midpoint of segment DQ. Point G is obtained by rotating point J by angle pi/12 about point D. Let C be the orthocenter of triangle JQD. Point P is the reflection of point K across point Q. Point V is the reflection of point C across line S. Point M is the reflection of point J across point V. Find the inradius of triangle GMP. Express your answer as a decimal number rounded to 2 decimal places.

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Let ZH = 10. Let A be the rotation of point Z by angle 11pi/12 around point H. Point L is the orthocenter of triangle ZHA. Let W be the midpoint of segment ZH. Point Q is the centroid of triangle ZHA. Circle S is the incircle of triangle ZHA. Point D is the midpoint of segment WL. Let C be the inversion of point Q in circle S. What is the distance between points D and C? Express your answer as a decimal number rounded to 2 decimal places.

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PZ = 10. Let U be the reflection of point Z across point P. Let O be the rotation of point U by angle pi/2 around point Z. Line A passes through points O and U. Let N be the line through point U perpendicular to line A. Let T be the line through point P perpendicular to line N. Let X be the reflection of line A across line T. Point F is the reflection of point P across line X. Let S be the midpoint of segment UP. What is the inradius of triangle OSF? Express your answer as a decimal number rounded to 2 decimal places.

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YD = 6. Let T be the line through points Y and D. Let R be the midpoint of segment DY. Point V is obtained by rotating point R by angle 5pi/6 about point Y. Point G is the midpoint of segment YV. Let K be the midpoint of segment YR. Line W is the perpendicular bisector of segment GD. Line A is the reflection of line W across point V. Let P be the reflection of line T across line A. Point Z is the reflection of point K across line P. Find the circumradius of triangle ZYG. Express your answer as a decimal number rounded to 2 decimal places.

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Line E passes through point F parallel to segment VI14. Point N is on line E with |NJ| = 1. Point C is obtained by rotating point J by angle 7pi/12 about point S. Point K is the reflection of point S across segment CV. Point H is the midpoint of segment SS. Let R be the midpoint of segment KH. What is the length of segment RS? Express your answer as a decimal number rounded to 2 decimal places.

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Let TY = 10. Point U is obtained by rotating point T by angle 3pi/4 about point Y. Point A is the midpoint of segment UY. Let K be the centroid of triangle YTU. Point E is the reflection of point A across point K. What is the length of segment UE? Express your answer as a decimal number rounded to 2 decimal places.

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Let RV = 2. Let J be the reflection of point R across point V. Let S be the rotation of point V by angle pi/2 around point J. Point U is the reflection of point V across point S. Point A is the circumcenter of triangle SUR. Point K is the reflection of point J across segment SU. What is the measure of angle RAK, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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GR = 1. Let E be the rotation of point R by angle pi/12 around point G. Point N is the midpoint of segment ER. Let M be the line through point E perpendicular to segment GE. Line J is the perpendicular bisector of segment GN. Let L be the midpoint of segment GE. Let A be the line through point R parallel to line M. Let T be the line through point L perpendicular to line J. Let H be the reflection of line T through point G. Point W is the intersection of lines A and H. Point U is the midpoint of segment RL. What is the circumradius of triangle GWU? Express your answer as a decimal number rounded to 2 decimal places.

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IV = 6. Let X be the rotation of point V by angle pi/12 around point I. Let Q be the midpoint of segment VI. Let U be the reflection of point V across point X. Point B is the midpoint of segment QU. Circle D is the circumcircle of triangle BVU. Let T be the inversion of point I in circle D. Let L be the incenter of triangle BVU. Point Y is the reflection of point T across point V. What is the distance between points L and Y? Express your answer as a decimal number rounded to 2 decimal places.

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Let PL = 1. Let U be the vector from point L to point P. Point R is the translation of point P by vector U. Let I be the rotation of point P by angle pi/6 around point R. Point A is the orthocenter of triangle PLI. Circle O is the circumcircle of triangle PLI. Let N be the inversion of point R in circle O. What is the distance between points A and N? Express your answer as a decimal number rounded to 2 decimal places.

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Let YB = 5. Point D is obtained by rotating point Y by angle 11pi/12 about point B. Let I be the angle bisector of angle YBD. Line M is the reflection of line I across point Y. Let T be the midpoint of segment BD. Point A is the reflection of point Y across line M. Let F be the circumcircle of triangle YBD. Let W be the inversion of point T in circle F. Find the distance between points A and W. Express your answer as a decimal number rounded to 2 decimal places.

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Let KW = 8. Let R be the midpoint of segment KW. Point Q is obtained by rotating point W by angle 7pi/12 about point R. Point L is the midpoint of segment KQ. Let I be the incenter of triangle QWK. What is the inradius of triangle LIR? Express your answer as a decimal number rounded to 2 decimal places.

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PL = 5. Point F is the reflection of point P across point L. Let M be the rotation of point L by angle pi/3 around point F. Let Y be the reflection of point P across point F. Point V is the midpoint of segment LF. Let W be the circumcircle of triangle FML. Let N be the inversion of point Y in circle W. Point A is the reflection of point V across segment FM. Find the measure of angle ALN, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let PZ = 3. Line Y is the perpendicular bisector of segment ZP. Point T lies on line Y at distance 2 from point Z. Point J is the midpoint of segment PZ. Point V is the reflection of point J across point T. Let I be the perpendicular bisector of segment ZV. Let W be the reflection of line I across line Y. Line W and segment ZV intersect at point Q. Find the length of segment PQ. Express your answer as a decimal number rounded to 2 decimal places.

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Let NY = 9. Let U be the rotation of point Y by angle 5pi/12 around point N. Let K be the reflection of point N across point Y. Let C be the reflection of point K across point N. Let V be the midpoint of segment UY. Let F be the midpoint of segment CV. What is the circumradius of triangle FKV? Express your answer as a decimal number rounded to 2 decimal places.

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RF = 7. Let S be the rotation of point R by angle 5pi/6 around point F. Point I is the reflection of point S across segment FR. Let H be the angle bisector of angle ISF. Point M is the midpoint of segment FR. Point C is the reflection of point M across line H. Let B be the midpoint of segment CF. What is the measure of angle BIR, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let LY = 11. Let D be the rotation of point Y by angle pi/12 around point L. Let F be the rotation of point L by angle pi/4 around point D. Line N is the angle bisector of angle YLF. Line W is the reflection of line N across point D. Point P is the centroid of triangle LYD. Point I is the reflection of point P across line W. What is the distance between points L and I? Express your answer as a decimal number rounded to 2 decimal places.

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Let WV = 3. Point Q is the reflection of point W across point V. Let K be the midpoint of segment QV. Point F is obtained by rotating point K by angle pi/4 about point Q. Point U is the midpoint of segment FK. Let R be the line through point U perpendicular to segment FW. Point N is the reflection of point Q across line R. Point S is the reflection of point N across segment UF. Find the circumradius of triangle UNS. Express your answer as a decimal number rounded to 2 decimal places.

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Let IQ = 6. Point V is obtained by rotating point I by angle pi/3 about point Q. Circle W is the incircle of triangle VQI. Let O be the midpoint of segment QI. Point A is the inversion of point I with respect to circle W. Line U is the perpendicular bisector of segment IA. Point C is the reflection of point Q across line U. Find the measure of angle OAC, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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NR = 2. Let V be the reflection of point R across point N. Let S be the rotation of point R by angle 11pi/12 around point V. Let D be the line through points V and S. Line H is the angle bisector of angle VNS. Let K be the rotation of point R by angle pi/4 around point S. Point E is the reflection of point K across line D. Let Z be the midpoint of segment VE. Point B is the reflection of point Z across line H. Find the inradius of triangle KBN. Express your answer as a decimal number rounded to 2 decimal places.

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FS = 11. Point L is obtained by rotating point S by angle 7pi/12 about point F. Point B is the reflection of point S across point F. Let K be the reflection of point S across point L. Point T is the incenter of triangle FBL. What is the length of segment TK? Express your answer as a decimal number rounded to 2 decimal places.

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QP = 6. Let N be the rotation of point P by angle 3pi/4 around point Q. Let E be the reflection of point Q across point N. Let J be the midpoint of segment PQ. Line I is the perpendicular bisector of segment NE. Point M is the reflection of point J across line I. Find the inradius of triangle NME. Express your answer as a decimal number rounded to 2 decimal places.

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AR = 8. Point U is the reflection of point R across point A. Let C be the rotation of point R by angle pi/12 around point A. Circle F is the circumcircle of triangle ACR. Let G be the reflection of point R across point U. Point S is the midpoint of segment CR. Let Q be the inversion of point S in circle F. Find the distance between points Q and G. Express your answer as a decimal number rounded to 2 decimal places.

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BP = 1. Point K is obtained by rotating point B by angle 7pi/12 about point P. Let H be the rotation of point P by angle pi/4 around point B. Let J be the circumcircle of triangle BHP. Point S is the inversion of point K with respect to circle J. Let L be the incenter of triangle BHP. What is the distance between points L and S? Express your answer as a decimal number rounded to 2 decimal places.

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Let PG = 8. Point I is obtained by rotating point P by angle pi/4 about point G. Let N be the incenter of triangle IGP. Line S passes through point I perpendicular to segment IG. Point J is the reflection of point N across line S. Point K is the reflection of point P across point J. What is the measure of angle NKI, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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SQ = 4. Point B is obtained by rotating point S by angle pi/2 about point Q. Let C be the vector from point B to point Q. Point J is the reflection of point S across point B. Let M be point S translated by vector C. Line H is the angle bisector of angle QMJ. Point E is the centroid of triangle BQS. Let Z be the midpoint of segment ES. Point X is the reflection of point J across line H. Let V be the reflection of point M across point X. What is the measure of angle VZJ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let TI = 1. Let V be the reflection of point I across point T. Let P be the rotation of point T by angle pi/4 around point I. Point K is the reflection of point P across point V. Let H be the circumcenter of triangle KIP. Let R be the reflection of point K across point H. Find the length of segment RH. Express your answer as a decimal number rounded to 2 decimal places.

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DG = 10. Let N be the rotation of point G by angle pi/2 around point D. Let I be the midpoint of segment DG. Let T be the rotation of point D by angle pi/4 around point I. Line Z is the perpendicular bisector of segment TN. Point J is the reflection of point G across line Z. Let S be the line through point T parallel to segment ID. Point V is the reflection of point J across line S. Point E is the midpoint of segment NV. Point B is the midpoint of segment ID. Find the measure of angle EBJ, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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RU = 6. Let T be the rotation of point U by angle 7pi/12 around point R. Let L be the rotation of point T by angle pi around point R. Let O be the perpendicular bisector of segment UR. Point Y is the circumcenter of triangle UTL. Line F is the reflection of line O across point L. Point X is the reflection of point Y across line F. Find the distance between points Y and X. Express your answer as a decimal number rounded to 2 decimal places.

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VY = 9. Point K is obtained by rotating point Y by angle pi/12 about point V. Let H be the midpoint of segment VY. Let C be the rotation of point K by angle pi/4 around point H. Point T is the midpoint of segment VC. Circle M is the incircle of triangle YVC. Let I be the circumcenter of triangle YVC. Point A is the inversion of point I with respect to circle M. What is the length of segment AT? Express your answer as a decimal number rounded to 2 decimal places.

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MU = 3. Let Z be the rotation of point M by angle pi/6 around point U. Point J is the reflection of point Z across point U. Point Y is the orthocenter of triangle UZM. Circle S is the circumcircle of triangle UZM. Line B passes through point Z perpendicular to segment UZ. Let A be the line through point J parallel to line B. Point P is the reflection of point U across line A. Let N be the line through point Y perpendicular to line A. Let C be the reflection of circle S across line N. Point Q is the inversion of point P with respect to circle C. What is the length of segment ZQ? Express your answer as a decimal number rounded to 2 decimal places.

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SB = 1. Line H passes through points B and S. Let C be the rotation of point B by angle 5pi/12 around point S. Point F is the midpoint of segment SB. Let K be the line through points C and F. Point W is the reflection of point B across line K. Point A is the reflection of point C across line H. Let P be the reflection of point A across point F. What is the measure of angle SPW, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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DG = 5. Let N be the rotation of point G by angle pi/3 around point D. Let W be the midpoint of segment DG. Point M is the reflection of point W across point G. Circle S is the incircle of triangle GND. Let A be the inversion of point N in circle S. What is the length of segment AM? Express your answer as a decimal number rounded to 2 decimal places.

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Let TE = 7. Point A is obtained by rotating point T by angle 5pi/12 about point E. Point J is the reflection of point A across point T. Point B is obtained by rotating point J by angle pi/6 about point T. Point C is the midpoint of segment EJ. Point U is the orthocenter of triangle JTE. Point Z is the circumcenter of triangle JTE. Point V is the reflection of point C across point U. Point O is the midpoint of segment BV. What is the circumradius of triangle ZJO? Express your answer as a decimal number rounded to 2 decimal places.

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TB = 3. Point C is obtained by rotating point B by angle 2pi/3 about point T. Line J is the angle bisector of angle TBC. Let O be the reflection of point C across point T. Point H is on line J with |HB| = 4. Point I is the reflection of point T across line J. Let G be the line through point I perpendicular to segment TO. Point A is the reflection of point T across line G. What is the measure of angle IHA, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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RJ = 3. Point D is obtained by rotating point R by angle pi/3 about point J. Point Z is the midpoint of segment RJ. Let N be the rotation of point D by angle 11pi/12 around point J. Line S passes through points D and N. Let T be the line through points Z and N. Let X be the reflection of line S across line T. Point H is the reflection of point J across line X. Point Y is the midpoint of segment DH. Point G is the reflection of point Y across line S. Point Q is the reflection of point G across line X. Point K is the midpoint of segment QZ. What is the measure of angle HKQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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RY = 2. Line I passes through point R perpendicular to segment YR. Point O is on line I with |OY| = 9. Line L passes through point Y parallel to line I. Point W is the reflection of point R across line L. Let S be the reflection of point O across point Y. Point B is on line L. Find the measure of angle SBW, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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KP = 11. Let Y be the midpoint of segment PK. Let Q be the rotation of point K by angle 7pi/12 around point Y. Let S be the midpoint of segment PQ. Point I is the reflection of point S across point P. Point L is the reflection of point S across segment QY. Find the length of segment IL. Express your answer as a decimal number rounded to 2 decimal places.

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Let ZX = 6. Point Q is the reflection of point Z across point X. Line J is the angle bisector of angle ZXQ. Point W is obtained by rotating point Q by angle 5pi/12 about point Z. Point E is the reflection of point Q across point W. Point U is obtained by rotating point Q by angle 7pi/12 about point X. Line M passes through point W parallel to line J. Let K be the reflection of line M through point U. Line K and segment QE intersect at point N. Point V is the reflection of point Z across point N. Find the measure of angle VUN, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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VH = 10. Let Y be the midpoint of segment HV. Let G be the rotation of point V by angle pi/3 around point H. Let K be the rotation of point Y by angle 5pi/12 around point V. Let B be the incenter of triangle HKY. Let U be the angle bisector of angle KGV. Point X is the reflection of point K across line U. What is the length of segment XB? Express your answer as a decimal number rounded to 2 decimal places.

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PO = 3. Point E is obtained by rotating point P by angle pi/6 about point O. Let I be the midpoint of segment EP. Let C be the midpoint of segment IE. Point D is the reflection of point C across point E. What is the inradius of triangle OID? Express your answer as a decimal number rounded to 2 decimal places.

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Let CV = 12. Let K be the perpendicular bisector of segment CV. Point L is on line K with |LV| = 10. Point J is the reflection of point C across point L. Point F is the reflection of point L across segment VJ. Find the inradius of triangle LFV. Express your answer as a decimal number rounded to 2 decimal places.

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SI = 6. Let W be the rotation of point I by angle pi around point S. Let J be the vector from point W to point I. Let O be the rotation of point W by angle pi/3 around point I. Point B is the translation of point S by vector J. Let E be the perpendicular bisector of segment IO. Let A be the reflection of line E through point S. Point K is the reflection of point O across point B. Let R be the line through point S perpendicular to segment SI. Line P is the reflection of line R across line A. Point H is the reflection of point O across line P. Find the measure of angle HSK, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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AF = 12. Let T be the reflection of point A across point F. Point Q is obtained by rotating point A by angle 11pi/12 about point F. Let B be the midpoint of segment QF. Point I is the reflection of point B across point T. Let L be the reflection of point I across point A. Let N be the midpoint of segment TL. Point R is the midpoint of segment NQ. Let P be the reflection of point I across point R. Let G be the reflection of point N across point P. Find the distance between points B and G. Express your answer as a decimal number rounded to 2 decimal places.

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PM = 6. Point V is obtained by rotating point M by angle 2pi/3 about point P. Point W is the reflection of point M across point V. Line J is the perpendicular bisector of segment WV. Point S is the reflection of point P across line J. Point I is the orthocenter of triangle SPV. Find the distance between points I and S. Express your answer as a decimal number rounded to 2 decimal places.

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NY = 10. Let B be the reflection of point Y across point N. Let E be the rotation of point N by angle pi/4 around point Y. Let J be the midpoint of segment EB. Let C be the midpoint of segment NE. Point R is the reflection of point C across segment YN. Let Z be the circumcenter of triangle NRJ. Find the distance between points Z and E. Express your answer as a decimal number rounded to 2 decimal places.

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QF = 11. Point Z is the reflection of point F across point Q. Let J be the rotation of point Z by angle 5pi/12 around point F. Point K is the reflection of point J across point Q. Let U be the midpoint of segment ZK. Point E is the reflection of point U across point Z. Find the circumradius of triangle EKF. 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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YP = 11. Let Q be the reflection of point Y across point P. Let N be the midpoint of segment YP. Let V be the rotation of point N by angle pi/6 around point P. Point J is the reflection of point N across point V. Line W is the angle bisector of angle YQV. Let L be the rotation of point Y by angle pi around point J. Point Z is on line W. Let S be the reflection of point L across point N. Let F be the midpoint of segment VS. What is the distance between points F and Q? Express your answer as a decimal number rounded to 2 decimal places.

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DG = 11. Let L be the line through points G and D. Point K is obtained by rotating point D by angle pi about point G. Let T be the rotation of point G by angle 2pi/3 around point K. Let V be the midpoint of segment GK. Point B is the reflection of point T across line L. Let E be the reflection of point B across point G. Point N is the midpoint of segment TE. Find the inradius of triangle VNB. Express your answer as a decimal number rounded to 2 decimal places.

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Let VH = 9. Let G be the line through points V and H. Let M be the line through point V perpendicular to line G. Let W be the midpoint of segment VH. Point Z is on line M with |ZW| = 6. Point T is on line G. Line A is the angle bisector of angle TWZ. Line D is the reflection of line A across point Z. Point B is the reflection of point H across line D. Point R is the reflection of point Z across line D. What is the inradius of triangle BHR? Express your answer as a decimal number rounded to 2 decimal places.

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IG = 12. Point Y is obtained by rotating point I by angle 5pi/6 about point G. Point Q is the midpoint of segment YG. Point H is the orthocenter of triangle QIG. Point F is the reflection of point H across segment GQ. Find the measure of angle YIF, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let NF = 9. Let L be the reflection of point N across point F. Let E be the reflection of point F across point L. Point U is obtained by rotating point N by angle 5pi/12 about point F. Let S be the rotation of point U by angle 5pi/12 around point L. Point O is the reflection of point S across segment EU. Point D is the midpoint of segment EL. Line A passes through point E perpendicular to segment EU. Line C is the reflection of line A across point D. Point Z is the reflection of point O across line C. Find the inradius of triangle UZF. Express your answer as a decimal number rounded to 2 decimal places.

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VC = 7. Point B is obtained by rotating point V by angle pi/4 about point C. Let Z be the rotation of point B by angle pi/3 around point C. Point J is the reflection of point V across point Z. Point F is the reflection of point Z across segment VB. Let T be the midpoint of segment VB. Let Y be the midpoint of segment JT. Let N be the reflection of point V across point Y. What is the measure of angle CTN, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let TE = 3. Point Q is the midpoint of segment TE. Let G be the rotation of point E by angle pi/2 around point T. Let F be the perpendicular bisector of segment QE. Point P is the reflection of point G across line F. Let R be the orthocenter of triangle GPE. Point Z is the centroid of triangle GPE. Point X is the reflection of point T across point Z. Find the length of segment RX. Express your answer as a decimal number rounded to 2 decimal places.

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OC = 5. Let I be the reflection of point O across point C. Let P be the rotation of point O by angle pi/3 around point I. Point W is the reflection of point I across point O. Line E is the perpendicular bisector of segment CP. Let Y be the reflection of point C across point P. Point B is the reflection of point O across line E. Find the measure of angle BWY, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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TP = 2. Line W is the perpendicular bisector of segment PT. Let G be the reflection of line W through point P. Point M is on line G with |MP| = 11. Let Z be the midpoint of segment MP. Line K is the perpendicular bisector of segment PZ. Point U is the midpoint of segment ZP. Line O passes through points T and M. Let D be the reflection of line K across line O. Point A is the reflection of point Z across line D. Point L is the midpoint of segment TZ. Find the circumradius of triangle AUL. Express your answer as a decimal number rounded to 2 decimal places.

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Line Q is the perpendicular bisector of segment OO. Point L is the reflection of point O across segment R46T. Point V is the reflection of point L across line Q. Let H be the circumcircle of triangle T. Point E is the inversion of point Z with respect to circle H. Point G is the reflection of point E across segment FR46. Let N be the midpoint of segment TV. Find the length of segment FN. Express your answer as a decimal number rounded to 2 decimal places.

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Let HF = 4. Let D be the rotation of point F by angle pi/2 around point H. Let Q be the centroid of triangle HDF. Let U be the incenter of triangle HDF. What is the inradius of triangle QUF? Express your answer as a decimal number rounded to 2 decimal places.

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QE = 12. Point P is obtained by rotating point E by angle 5pi/6 about point Q. Point S is the reflection of point Q across point E. Let V be the orthocenter of triangle PQE. Point W is the reflection of point E across point S. What is the length of segment WV? Express your answer as a decimal number rounded to 2 decimal places.

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Let XA = 2. Point J is the midpoint of segment AX. Let G be the rotation of point A by angle 5pi/12 around point J. Let M be the rotation of point G by angle pi/12 around point A. Let B be the centroid of triangle GJM. Let Q be the midpoint of segment MJ. Point R is the centroid of triangle AQB. Let Z be the incircle of triangle AQB. Point F is the inversion of point Q with respect to circle Z. What is the distance between points R and F? Express your answer as a decimal number rounded to 2 decimal places.

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Let VZ = 7. Let W be the rotation of point Z by angle 5pi/6 around point V. Point A is the midpoint of segment ZW. Line Q is the perpendicular bisector of segment VZ. Let U be the line through point W perpendicular to line Q. Point S is the reflection of point V across line U. Find the circumradius of triangle AWS. Express your answer as a decimal number rounded to 2 decimal places.

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Let TL = 1. Point R is obtained by rotating point L by angle 5pi/12 about point T. Point J is the midpoint of segment LT. Let N be the reflection of point T across point R. Point W is the reflection of point N across point J. Find the distance between points W and N. Express your answer as a decimal number rounded to 2 decimal places.

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Let AZ = 2. Let B be the reflection of point Z across point A. Let E be the rotation of point Z by angle 5pi/12 around point A. Point H is the midpoint of segment BE. Point F is the incenter of triangle ZBE. Point X is the reflection of point F across segment ZB. Point W is the reflection of point H across point B. What is the measure of angle XAW, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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QC = 5. Point X is obtained by rotating point Q by angle pi/6 about point C. Point Z is the reflection of point C across point Q. Point O is the reflection of point Q across point Z. Let W be the circumcenter of triangle QXC. Let M be the midpoint of segment XC. Point T is the reflection of point O across point W. Find the measure of angle TQM, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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Let EK = 11. Let D be the midpoint of segment EK. Let G be the rotation of point D by angle pi/12 around point E. Line W is the perpendicular bisector of segment GE. Line S is the perpendicular bisector of segment KE. Let P be the line through point E parallel to line S. Let C be the reflection of line W through point D. Line O passes through point K parallel to line P. Point V is the reflection of point G across segment KE. Point I is the intersection of lines C and O. What is the circumradius of triangle VID? Express your answer as a decimal number rounded to 2 decimal places.

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Let AV = 11. Line P is the perpendicular bisector of segment VA. Point O lies on line P at distance 1 from point A. Let X be the reflection of point A across point O. Let L be the vector from point X to point A. Point Y is the translation of point O by vector L. Line Z passes through points A and O. Let D be the midpoint of segment VY. Point W is the reflection of point V across line Z. Let Q be the incenter of triangle AWD. What is the length of segment QY? Express your answer as a decimal number rounded to 2 decimal places.

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Let QY = 3. Point Z is the midpoint of segment QY. Point B is obtained by rotating point Y by angle pi/4 about point Q. Let K be the centroid of triangle YBQ. Point I is the reflection of point Z across segment YB. Point D is the midpoint of segment KI. What is the measure of angle BDI, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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HR = 1. Let W be the rotation of point R by angle 5pi/12 around point H. Point Q is obtained by rotating point W by angle pi/3 about point H. Point M is the reflection of point R across point H. Point S is the reflection of point Q across segment RH. Let P be the midpoint of segment WR. Find the inradius of triangle MSP. Express your answer as a decimal number rounded to 2 decimal places.

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MA = 3. Point P is the midpoint of segment AM. Let N be the rotation of point P by angle 11pi/12 around point M. Line R is the perpendicular bisector of segment NM. Point E is the incenter of triangle MPN. Let B be the line through point P parallel to line R. Point U is the reflection of point A across line R. Let S be the line through point M parallel to segment EN. Point C is the intersection of lines S and B. What is the inradius of triangle UEC? Express your answer as a decimal number rounded to 2 decimal places.

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Let VC = 4. Line G is the perpendicular bisector of segment VC. Line J passes through point V parallel to line G. Point Y is on line J with |YV| = 1. Line H is the reflection of line G across line J. Let L be the midpoint of segment YC. Let A be the midpoint of segment VC. Point I is the reflection of point L across line H. What is the distance between points I and A? Express your answer as a decimal number rounded to 2 decimal places.

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Let XU = 9. Let V be the rotation of point X by angle 2pi/3 around point U. Point A is obtained by rotating point V by angle 11pi/12 about point X. Let Z be the reflection of point U across point V. Let O be the orthocenter of triangle AXZ. Point B is the circumcenter of triangle AXZ. Find the measure of angle XBO, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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DE = 11. Point W is obtained by rotating point E by angle pi/12 about point D. Point M is the reflection of point E across point D. Let S be the rotation of point W by angle 3pi/4 around point M. Point P is the reflection of point S across point D. Point Z is the reflection of point P across segment ES. Find the inradius of triangle ZSM. Express your answer as a decimal number rounded to 2 decimal places.

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AT = 11. Point U is the reflection of point T across point A. Point L is obtained by rotating point T by angle 2pi/3 about point A. Point I is the midpoint of segment LT. Point X is the reflection of point I across point U. Point G is the midpoint of segment XT. Point O is the circumcenter of triangle TUX. Let V be the incircle of triangle GIL. Let S be the inversion of point U in circle V. Find the measure of angle OXS, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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GJ = 11. Let H be the rotation of point G by angle 7pi/12 around point J. Point R is the midpoint of segment GH. Let M be the midpoint of segment HR. Point U is obtained by rotating point J by angle pi/12 about point G. Point T is obtained by rotating point U by angle pi/3 about point G. Line N is the perpendicular bisector of segment TR. Point Y is the midpoint of segment UM. Point A is the reflection of point T across segment UM. Line Z is the reflection of line N across point A. Point W is the reflection of point J across line Z. Let S be the reflection of point H across point W. What is the measure of angle YJS, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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KG = 2. Let I be the midpoint of segment GK. Point O is obtained by rotating point G by angle 3pi/4 about point I. Y is placed 1 units away from G. Let U be the circumcenter of triangle GIO. Point J is the midpoint of segment IG. Let L be the midpoint of segment IO. Point A is the reflection of point L across segment UJ. Find the circumradius of triangle UKA. Express your answer as a decimal number rounded to 2 decimal places.

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UG = 6. Let H be the rotation of point G by angle 11pi/12 around point U. Point P is the reflection of point H across point G. Point S is the midpoint of segment GU. Let F be the midpoint of segment HG. Point Q is the centroid of triangle GSH. Point O is the reflection of point Q across point P. What is the distance between points O and F? Express your answer as a decimal number rounded to 2 decimal places.

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XS = 2. Let V be the vector from point X to point S. Point Z is the reflection of point X across point S. Point A is the translation of point Z by vector V. Point Y is obtained by rotating point S by angle 3pi/4 about point A. Let B be the circumcenter of triangle ZXY. Line H is the perpendicular bisector of segment XY. Line K is the reflection of line H across point B. Point T is the midpoint of segment XY. Point L is the intersection of line K and segment YZ. What is the distance between points T and L? Express your answer as a decimal number rounded to 2 decimal places.

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Let LB = 6. Let W be the rotation of point B by angle 11pi/12 around point L. Point P is the midpoint of segment LW. Point O is the reflection of point P across segment WB. Point T is the reflection of point O across point P. What is the distance between points T and L? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let YG = 7. Let B be the rotation of point Y by angle pi/3 around point G. Let K be the midpoint of segment YG. Point R is the reflection of point B across segment GY. Let X be the vector from point R to point K. Point E is the reflection of point G across point B. Point S is the translation of point G by vector X. Let J be the midpoint of segment BG. Line H passes through point J parallel to segment GY. Line U passes through points Y and R. Lines U and H intersect at point M. Find the circumradius of triangle ESM. Express your answer as a decimal number rounded to 2 decimal places.

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Let RQ = 1. Point E is obtained by rotating point Q by angle 5pi/12 about point R. Let I be the midpoint of segment RQ. Point O is the centroid of triangle RQE. Circle V is the circumcircle of triangle RQE. Point B is the inversion of point I with respect to circle V. Find the distance between points O and B. Express your answer as a decimal number rounded to 2 decimal places.

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Let ZQ = 4. Point E is the reflection of point Q across point Z. Let R be the rotation of point Z by angle 7pi/12 around point E. Let H be the rotation of point E by angle 5pi/12 around point Z. Point N is the reflection of point R across point H. Let K be the midpoint of segment QZ. Let L be the reflection of point K across point N. Let O be the midpoint of segment NH. What is the inradius of triangle KOL? Express your answer as a decimal number rounded to 2 decimal places.

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YB = 9. Let Q be the rotation of point B by angle 5pi/6 around point Y. Let H be the midpoint of segment YQ. Let E be the orthocenter of triangle YBQ. Point T is the midpoint of segment BQ. Point F is the reflection of point T across segment YB. What is the measure of angle HFE, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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Let KJ = 4. Point W is obtained by rotating point J by angle 5pi/12 about point K. Line Y is the perpendicular bisector of segment WJ. Let I be the midpoint of segment WK. Point L is the midpoint of segment JW. Point M is the reflection of point I across line Y. Find the distance between points M and L. Express your answer as a decimal number rounded to 2 decimal places.

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JF = 7. Line X is the perpendicular bisector of segment JF. Let O be the reflection of line X through point J. Let Y be the rotation of point J by angle 5pi/6 around point F. Let E be the orthocenter of triangle FJY. Point T is the midpoint of segment JF. Line R is the reflection of line O across point T. Segment YT and line R intersect at point Z. Find the length of segment EZ. Express your answer as a decimal number rounded to 2 decimal places.

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HW = 3. Let E be the rotation of point H by angle 5pi/6 around point W. Let Y be the midpoint of segment WE. Line U is the perpendicular bisector of segment YE. Point P is the reflection of point H across line U. Point F is the circumcenter of triangle PHE. 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 SF = 3. Point K is obtained by rotating point F by angle 5pi/12 about point S. Circle Y is the circumcircle of triangle FSK. Circle H is the reflection of circle Y across point S. Point U is the inversion of point F with respect to circle H. Point J is the orthocenter of triangle FSK. Line A passes through point J parallel to segment KF. Point V is the intersection of segment SK and line A. Point R is the centroid of triangle FSK. Line A and segment UR intersect at point G. Point P is the incenter of triangle VUR. What is the distance between points P and G? Express your answer as a decimal number rounded to 2 decimal places.

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PJ = 4. Let V be the rotation of point P by angle pi/3 around point J. Point X is the midpoint of segment JV. Circle H is the circumcircle of triangle VXP. Point U is the inversion of point J with respect to circle H. Point D is the centroid of triangle VXP. What is the circumradius of triangle UVD? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
IM = 11. Let W be the rotation of point M by angle 11pi/12 around point I. Point D lies on segment IW. Line X is the perpendicular bisector of segment IW. Point V is the reflection of point M across line X. Find the measure of angle VMD, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
OJ = 7. Let A be the reflection of point O across point J. Let Y be the perpendicular bisector of segment JA. Point E lies on line Y at distance 3 from point A. Point T is the reflection of point O across line Y. Point G is the circumcenter of triangle TEJ. Point S is the reflection of point G across line Y. Find the distance between points S and T. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
YC = 10. Point O is obtained by rotating point C by angle pi/12 about point Y. Let K be the reflection of point C across point Y. Let E be the midpoint of segment KY. Point M is the reflection of point K across segment OY. What is the measure of angle EMO, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
FP = 2. Point R is obtained by rotating point P by angle 3pi/4 about point F. Let O be the centroid of triangle FPR. Point S is the midpoint of segment RO. Let C be the orthocenter of triangle FPR. Line J is the angle bisector of angle FOC. Point K is the reflection of point S across line J. What is the measure of angle CPK, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
AO = 8. Point S is obtained by rotating point A by angle 5pi/12 about point O. Line G passes through points A and S. Point J is the reflection of point O across line G. Point Y is the midpoint of segment SO. Let N be the perpendicular bisector of segment SY. Point Q is the reflection of point A across line N. Point U is obtained by rotating point Q by angle pi about point O. Let F be the perpendicular bisector of segment AY. Point B is the reflection of point U across point J. Point I is the reflection of point O across line F. What is the inradius of triangle BJI? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
YH = 2. Point W is the midpoint of segment HY. Let L be the reflection of point W across point H. Point E is obtained by rotating point H by angle 11pi/12 about point L. Let U be the rotation of point L by angle 2pi/3 around point W. Let K be the reflection of point W across point L. Point V is obtained by rotating point E by angle 7pi/12 about point H. Point D is the midpoint of segment YU. Point I is obtained by rotating point K by angle pi/6 about point V. Let Q be the reflection of point L across point Y. Let F be the midpoint of segment DI. What is the circumradius of triangle EFQ? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
Let BX = 11. Let T be the rotation of point B by angle 3pi/4 around point X. Let P be the rotation of point B by angle 5pi/12 around point X. Let Y be the midpoint of segment TX. Circle N is the incircle of triangle PBY. Let W be the orthocenter of triangle PBY. Let E be the inversion of point X in circle N. Find the circumradius of triangle ETW. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
KD = 8. Let P be the rotation of point K by angle 3pi/4 around point D. Line V is the perpendicular bisector of segment KP. Point L is the midpoint of segment PD. Let U be the reflection of line V through point P. Point R is the midpoint of segment PL. Point M is the reflection of point R across line U. What is the measure of angle MLR, in radians? Express your answer as a decimal number rounded to 2 decimal places.

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[User]
QM = 1. Let C be the rotation of point M by angle pi/12 around point Q. Point A is the reflection of point C across point M. Let O be the angle bisector of angle AQC. Let B be the midpoint of segment CM. Point L is the reflection of point B across line O. Let U be the circumcenter of triangle CBQ. Find the distance between points L and U. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
UY = 11. Point Z is the reflection of point Y across point U. Let H be the vector from point Y to point Z. Point M is the midpoint of segment ZU. Let E be the rotation of point Z by angle pi/6 around point U. Let K be point Z translated by vector H. Let G be the rotation of point K by angle pi/12 around point M. Let J be the reflection of point G across point E. Point C is the reflection of point J across segment ME. Find the inradius of triangle KJC. Express your answer as a decimal number rounded to 2 decimal places.

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[User]
FB = 2. Let T be the midpoint of segment BF. Let Z be the reflection of point T across point F. Point O is obtained by rotating point T by angle pi/6 about point Z. Point H is the reflection of point T across segment FO. Line L is the angle bisector of angle TBH. Point C is the reflection of point O across line L. Find the measure of angle CFH, in radians. Express your answer as a decimal number rounded to 2 decimal places.

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