messages listlengths 1 1 | ground_truth stringlengths 3 7 | dataset stringclasses 1 value |
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[
{
"content": "OQ = 12. Point E is obtained by rotating point Q by angle 5pi/6 about point O. Point I is the midpoint of segment QO. Let X be the line through point E perpendicular to segment EO. Point J is the reflection of point Q across line X. What is the circumradius of triangle IEJ? Express your answer as ... | 47.15 | MATH |
[
{
"content": "GC = 6. Let Q be the midpoint of segment GC. Let B be the rotation of point Q by angle pi/4 around point G. Line X is the perpendicular bisector of segment BG. Let D be the midpoint of segment BQ. Line R passes through point G parallel to line X. Point E is the midpoint of segment DB. Point W is t... | 5.15 | MATH |
[
{
"content": "ID = 11. Point R is the reflection of point D across point I. Let F be the rotation of point D by angle pi/4 around point I. Point Z is the reflection of point I across point F. Let J be the centroid of triangle RFZ. Point H is the reflection of point R across segment FZ. Find the length of segmen... | 19.22 | MATH |
[
{
"content": "IS = 3. Point F is obtained by rotating point I by angle 5pi/12 about point S. B is constructed so that FB = 3. Point V is the reflection of point S across segment IF. Let Q be the midpoint of segment IS. Let Y be the line through points B and V. Point K is the reflection of point Q across line Y.... | 3.69 | MATH |
[
{
"content": "IU = 6. Let O be the rotation of point U by angle pi/6 around point I. Point H is the midpoint of segment OU. Point M is the centroid of triangle OIH. Point S is the incenter of triangle OIH. What is the distance between points M and S? Express your answer as a decimal number rounded to 2 decimal ... | 1.27 | MATH |
[
{
"content": "Let TC = 3. Point E is obtained by rotating point T by angle pi/2 about point C. Point J is the reflection of point E across point C. Let H be the circumcenter of triangle TJE. Let L be the incenter of triangle TJE. Point A is the reflection of point L across point H. Find the distance between poi... | 2.24 | MATH |
[
{
"content": "HE = 1. Let P be the rotation of point H by angle 5pi/12 around point E. Point I is the reflection of point E across point P. Point B is the midpoint of segment PH. Point X is the reflection of point I across point B. What is the inradius of triangle PBX? Express your answer as a decimal number ro... | 0.12 | MATH |
[
{
"content": "AR = 5. Point C is obtained by rotating point A by angle 5pi/12 about point R. Let H be the perpendicular bisector of segment CR. Point G is the midpoint of segment AC. Point J is the reflection of point A across line H. Let V be the midpoint of segment JG. Point Y is the reflection of point V acr... | 2.5 | MATH |
[
{
"content": "AL = 3. Let R be the rotation of point L by angle pi/4 around point A. Let Q be the circumcenter of triangle ALR. Line Z passes through point Q parallel to segment AL. Segment LR and line Z intersect at point Y. Let F be the midpoint of segment LR. Find the distance between points Y and F. Express... | 0.38 | MATH |
[
{
"content": "BQ = 7. Point H is the midpoint of segment BQ. Point V is obtained by rotating point B by angle pi/6 about point Q. Line E passes through point H perpendicular to segment HB. Point D is the reflection of point V across segment HB. Point X is the reflection of point D across line E. Find the inradi... | 1.27 | MATH |
[
{
"content": "Let DR = 6. Let J be the rotation of point R by angle pi/3 around point D. Let N be the incenter of triangle DRJ. Let Z be the midpoint of segment NR. Let M be the midpoint of segment RJ. What is the measure of angle ZMN, in radians? Express your answer as a decimal number rounded to 2 decimal pla... | 1.05 | MATH |
[
{
"content": "BR = 8. Let X be the reflection of point R across point B. Let Y be the rotation of point B by angle 7pi/12 around point X. Let F be the reflection of point B across point X. Circle A is the circumcircle of triangle RBY. Point D is the reflection of point Y across segment RB. Let V be the inversio... | 2.05 | MATH |
[
{
"content": "Let UB = 10. Point D is obtained by rotating point U by angle 11pi/12 about point B. Let K be the angle bisector of angle BUD. Line W passes through point U perpendicular to line K. Point E is the reflection of point B across line W. Find the inradius of triangle EDB. Express your answer as a deci... | 0.65 | MATH |
[
{
"content": "UH = 9. Point J is obtained by rotating point H by angle 3pi/4 about point U. Let F be the angle bisector of angle UHJ. Point M is the reflection of point J across line F. Point N is the midpoint of segment MH. Let P be the reflection of point H across point U. Point K is the centroid of triangle ... | 7.42 | MATH |
[
{
"content": "CW = 12. Point O is obtained by rotating point W by angle 5pi/6 about point C. Point T is the midpoint of segment CO. Point K is the midpoint of segment CW. Point X is the reflection of point C across point W. Find the circumradius of triangle TXK. Express your answer as a decimal number rounded t... | 56.7 | MATH |
[
{
"content": "OD = 1. Let S be the midpoint of segment OD. Point K is obtained by rotating point D by angle pi/6 about point S. Let I be the midpoint of segment KO. Line F passes through points O and S. Let V be the reflection of point D across point I. Point B is the reflection of point V across line F. Find t... | 0.03 | MATH |
[
{
"content": "Let QH = 4. Let O be the rotation of point H by angle 5pi/6 around point Q. Let J be the midpoint of segment OQ. Line V is the angle bisector of angle OJH. Point L is the midpoint of segment QH. Point E is the reflection of point L across line V. Find the measure of angle LEJ, in radians. Express ... | 1.28 | MATH |
[
{
"content": "Let SF = 7. Line P passes through point S perpendicular to segment SF. Let X be the rotation of point S by angle 5pi/6 around point F. Let D be the midpoint of segment XS. Point O is the reflection of point D across line P. Let L be the centroid of triangle SDO. What is the distance between points... | 6.56 | MATH |
[
{
"content": "IV = 11. Point M is obtained by rotating point I by angle 3pi/4 about point V. Let S be the reflection of point M across point V. Point F is the midpoint of segment MV. Line Y is the angle bisector of angle SIV. Let H be the rotation of point F by angle pi around point I. Line W is the perpendicul... | 1012.45 | MATH |
[
{
"content": "Let RO = 5. Let Z be the perpendicular bisector of segment RO. Point T lies on line Z at distance 8 from point R. Point E is obtained by rotating point R by angle 7pi/12 about point T. Let V be the orthocenter of triangle OER. Point U is the midpoint of segment ER. Let F be the midpoint of segment... | 8.38 | MATH |
[
{
"content": "HE = 8. Point V is obtained by rotating point E by angle pi/3 about point H. Point I is the midpoint of segment EV. Point P is the centroid of triangle VHE. Let B be the midpoint of segment EP. Find the length of segment BI. Express your answer as a decimal number rounded to 2 decimal places.\n\nP... | 2.31 | MATH |
[
{
"content": "PA = 2. Point S is obtained by rotating point P by angle pi/6 about point A. Line L is the perpendicular bisector of segment SA. Let C be the midpoint of segment AS. Point W is the reflection of point P across line L. Let I be the reflection of point W across point S. What is the circumradius of t... | 2.91 | MATH |
[
{
"content": "Let ID = 6. Let N be the rotation of point I by angle 11pi/12 around point D. Line K is the perpendicular bisector of segment ID. Point A is obtained by rotating point N by angle pi about point I. Let U be the line through point A parallel to line K. Let Q be the rotation of point N by angle 11pi/... | 1.77 | MATH |
[
{
"content": "Let AB = 11. Let D be the rotation of point B by angle pi/12 around point A. Point S is the reflection of point A across point D. Let F be the orthocenter of triangle BAS. Circle J is the circumcircle of triangle BAS. Point Q is the inversion of point F with respect to circle J. What is the distan... | 54.01 | MATH |
[
{
"content": "ZK = 10. Point H is obtained by rotating point Z by angle 5pi/12 about point K. Let W be the midpoint of segment ZH. Let F be the midpoint of segment ZK. Point C is the reflection of point W across point Z. Line G is the perpendicular bisector of segment KH. Point Q is the reflection of point W ac... | 6.03 | MATH |
[
{
"content": "XD = 7. Let Q be the midpoint of segment DX. Point O is obtained by rotating point Q by angle 2pi/3 about point X. Point Z is the midpoint of segment DQ. Point A is the reflection of point Q across point O. Let G be the midpoint of segment QZ. Point Y is the circumcenter of triangle AZQ. What is t... | 4.3 | MATH |
[
{
"content": "GF = 2. Let R be the rotation of point G by angle 11pi/12 around point F. Circle E is the incircle of triangle GRF. Let U be the midpoint of segment RF. Point P is the inversion of point U with respect to circle E. What is the length of segment PR? Express your answer as a decimal number rounded t... | 1.97 | MATH |
[
{
"content": "HS = 8. Let Q be the rotation of point H by angle pi/6 around point S. Point B is the midpoint of segment HS. Point X is the midpoint of segment QS. Point P is the midpoint of segment XQ. Find the measure of angle BPH, in radians. Express your answer as a decimal number rounded to 2 decimal places... | 6.0 | MATH |
[
{
"content": "KX = 7. Let N be the reflection of point X across point K. Point L is obtained by rotating point K by angle 3pi/4 about point N. Let H be the midpoint of segment KL. Let R be the angle bisector of angle HLX. Point V is the reflection of point K across line R. What is the inradius of triangle HVN? ... | 1.0 | MATH |
[
{
"content": "YD = 11. Let V be the rotation of point Y by angle pi/12 around point D. Point N is the midpoint of segment VD. Let P be the orthocenter of triangle VNY. Let H be the centroid of triangle VNY. What is the measure of angle NHP, in radians? Express your answer as a decimal number rounded to 2 decima... | 1.24 | MATH |
[
{
"content": "Let UI = 3. Point V is obtained by rotating point U by angle 7pi/12 about point I. Point A is obtained by rotating point I by angle pi/2 about point U. Let X be the circumcircle of triangle IAV. Let F be the incenter of triangle IAV. Let Q be the inversion of point F in circle X. What is the lengt... | 4.44 | MATH |
[
{
"content": "Let AE = 6. Point X is obtained by rotating point E by angle pi/2 about point A. Let V be the rotation of point X by angle 3pi/4 around point A. Point W is the reflection of point V across point E. Point C is the reflection of point V across point A. Let R be the midpoint of segment XV. Find the c... | 19.97 | MATH |
[
{
"content": "Let WD = 3. Line G passes through points D and W. Point T is obtained by rotating point D by angle pi/4 about point W. Let N be the rotation of point D by angle 2pi/3 around point T. Let M be the perpendicular bisector of segment TN. Let K be the reflection of line M across line G. Let B be the re... | 4.45 | MATH |
[
{
"content": "Let ZQ = 3. Point E is obtained by rotating point Q by angle 5pi/12 about point Z. Point I is obtained by rotating point Q by angle 3pi/4 about point E. Let P be the reflection of point E across point Q. Let G be the centroid of triangle EZI. What is the circumradius of triangle PIG? Express your ... | 7.02 | MATH |
[
{
"content": "UW = 12. Point D is the midpoint of segment WU. Point Y is the reflection of point D across point U. Point S is obtained by rotating point D by angle pi/6 about point Y. Let G be the reflection of point W across point S. Point T is obtained by rotating point U by angle pi/4 about point W. Let X be... | 1.05 | MATH |
[
{
"content": "Let SL = 4. Let Y be the rotation of point S by angle 7pi/12 around point L. Point W is obtained by rotating point Y by angle 11pi/12 about point S. Let Z be the midpoint of segment YS. Let X be the reflection of point W across point Z. Point P is the reflection of point W across point Y. What is ... | 6.35 | MATH |
[
{
"content": "Let ZQ = 6. Let V be the rotation of point Z by angle 7pi/12 around point Q. Point Y is the centroid of triangle QZV. Point C is the midpoint of segment ZY. Point F is the reflection of point Y across segment QZ. Let A be the midpoint of segment VF. Find the distance between points A and C. Expres... | 3.9 | MATH |
[
{
"content": "Let EA = 5. Let Z be the midpoint of segment EA. Let P be the reflection of point Z across point E. Let H be the rotation of point A by angle 3pi/4 around point P. Let S be the reflection of point H across point P. Point N is the reflection of point Z across segment HS. Line Q is the perpendicular... | 8.78 | MATH |
[
{
"content": "DL = 4. Point M is obtained by rotating point L by angle 2pi/3 about point D. Let G be the midpoint of segment DL. Point J is the midpoint of segment DM. Point Z is the centroid of triangle LJG. Find the length of segment DZ. Express your answer as a decimal number rounded to 2 decimal places.\n\n... | 1.76 | MATH |
[
{
"content": "Let LD = 6. Let O be the line through points L and D. Let R be the rotation of point D by angle 3pi/4 around point L. Line E is the perpendicular bisector of segment RL. Let X be the reflection of line E through point L. Lines X and O intersect at point C. Let K be the midpoint of segment CD. Let ... | 5.25 | MATH |
[
{
"content": "Let QS = 7. Point D is obtained by rotating point Q by angle pi/2 about point S. Circle T is the circumcircle of triangle DSQ. Let I be the angle bisector of angle SDQ. Point R is the reflection of point Q across line I. Let F be the incenter of triangle DSQ. Let Z be the inversion of point F in c... | 5.36 | MATH |
[
{
"content": "HC = 3. Point B is obtained by rotating point C by angle 5pi/12 about point H. Point U is obtained by rotating point H by angle pi/12 about point B. Line F passes through point H parallel to segment UC. Point O is the reflection of point C across line F. Point P is the reflection of point B across... | 4.35 | MATH |
[
{
"content": "IK = 5. Let W be the midpoint of segment IK. Point R is the midpoint of segment WI. Point E is obtained by rotating point R by angle 5pi/6 about point I. Point Z is the reflection of point E across point W. Circle D is the incircle of triangle ZKE. Let G be the perpendicular bisector of segment KE... | 0.38 | MATH |
[
{
"content": "LX = 4. Let N be the rotation of point X by angle 5pi/12 around point L. Point T is the reflection of point N across point L. Point R is obtained by rotating point T by angle 5pi/6 about point L. Point Y is the midpoint of segment TX. Let K be the angle bisector of angle NTY. Point D is the reflec... | 2.06 | MATH |
[
{
"content": "Let GJ = 7. Point C is the reflection of point J across point G. Let N be the rotation of point G by angle pi/3 around point J. Let W be the reflection of point N across point J. Let F be the angle bisector of angle JWC. Line P is the perpendicular bisector of segment JG. Let L be the incircle of ... | 1.62 | MATH |
[
{
"content": "Let VF = 5. Line G passes through points V and F. Line D passes through point V perpendicular to line G. Point S lies on line D at distance 2 from point F. Point A is the midpoint of segment SF. Line R is the angle bisector of angle SVA. Point U is the midpoint of segment SV. Point I is the reflec... | 4.78 | MATH |
[
{
"content": "Let WE = 3. Let F be the rotation of point W by angle 2pi/3 around point E. Let A be the reflection of point F across point W. Let M be the reflection of point F across point A. Point I is the reflection of point F across segment WE. Find the length of segment IM. Express your answer as a decimal ... | 18.73 | MATH |
[
{
"content": "Let QT = 12. Point O is the reflection of point Q across point T. Line S is the angle bisector of angle OTQ. Point C is the reflection of point T across point O. Point U lies on line S at distance 6 from point C. Line N is the angle bisector of angle OQU. Line I is the reflection of line S across ... | 4.18 | MATH |
[
{
"content": "Let RC = 8. Let RW = 12. Point J is the reflection of point W across point C. Let Z be the midpoint of segment JR. Let S be the centroid of triangle RCJ. What is the distance between points S and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex... | 2.0 | MATH |
[
{
"content": "LQ = 4. Point U is the midpoint of segment LQ. Point X is obtained by rotating point Q by angle 2pi/3 about point L. Point W is obtained by rotating point U by angle pi/4 about point L. Let O be the reflection of point W across point X. Point Z is the reflection of point L across point O. Point R ... | 0.2 | MATH |
[
{
"content": "FE = 10. Point A is obtained by rotating point E by angle 11pi/12 about point F. Let U be the midpoint of segment EF. Let L be the reflection of point A across point F. Let T be the incenter of triangle EAF. What is the measure of angle TUL, in radians? Express your answer as a decimal number roun... | 4.85 | MATH |
[
{
"content": "Let LY = 4. Point J is obtained by rotating point L by angle pi about point Y. Let M be the line through points J and L. Let O be the rotation of point J by angle 5pi/12 around point L. Let H be the rotation of point J by angle pi/12 around point Y. Line C passes through point O perpendicular to l... | 51.42 | MATH |
[
{
"content": "CE = 6. Line U is the perpendicular bisector of segment EC. Let Q be the reflection of line U through point E. Point N is on line Q with |NE| = 12. Point H is the reflection of point E across line Q. Let V be the midpoint of segment HN. Point P is the circumcenter of triangle CNV. What is the leng... | 8.85 | MATH |
[
{
"content": "Let CO = 7. Let U be the reflection of point O across point C. Let G be the rotation of point C by angle 3pi/4 around point O. Let B be the perpendicular bisector of segment OC. Line D is the reflection of line B across point C. Let S be the circumcenter of triangle UCG. Line D and segment GU inte... | 2.88 | MATH |
[
{
"content": "LT = 9. Let Y be the rotation of point L by angle 3pi/4 around point T. Let F be the midpoint of segment YT. Let D be the midpoint of segment FL. Let H be the orthocenter of triangle LYF. Let P be the reflection of point H across point D. What is the measure of angle PHL, in radians? Express your ... | 177.04 | MATH |
[
{
"content": "PZ = 7. Point I is obtained by rotating point Z by angle pi/2 about point P. Let O be the line through point I perpendicular to segment PI. Let B be the incircle of triangle PIZ. Point T is the reflection of point Z across line O. Let W be the inversion of point T in circle B. Let U be the midpoin... | 1.75 | MATH |
[
{
"content": "SB = 10. Let Q be the reflection of point B across point S. Let X be the rotation of point B by angle 7pi/12 around point Q. Let J be the midpoint of segment SQ. Point R is obtained by rotating point S by angle 5pi/12 about point Q. Let D be the midpoint of segment JX. Let H be the circumcenter of... | 19.34 | MATH |
[
{
"content": "PC = 9. Let Y be the midpoint of segment PC. Let Z be the rotation of point Y by angle 7pi/12 around point P. Let H be the midpoint of segment YZ. Let M be the midpoint of segment CZ. Let G be the line through point P perpendicular to segment PC. Point D is the reflection of point P across point M... | 7.25 | MATH |
[
{
"content": "Let MY = 4. Point W is obtained by rotating point Y by angle pi/3 about point M. Line U is the perpendicular bisector of segment WY. Line X is the reflection of line U across point W. Point R is the reflection of point W across line X. What is the inradius of triangle RMW? Express your answer as a... | 0.93 | MATH |
[
{
"content": "YX = 1. Let D be the rotation of point X by angle 5pi/12 around point Y. Point W is the midpoint of segment YD. Point E is obtained by rotating point X by angle 11pi/12 about point Y. Point I is the midpoint of segment XW. Line F passes through points E and I. Point K is the reflection of point W ... | 0.35 | MATH |
[
{
"content": "YR = 7. Let V be the rotation of point Y by angle 3pi/4 around point R. Let W be the rotation of point R by angle 2pi/3 around point Y. Point H is the circumcenter of triangle RVW. Point J is the reflection of point Y across segment VW. Let F be the centroid of triangle HJV. What is the distance b... | 5.82 | MATH |
[
{
"content": "NA = 4. Point X is the midpoint of segment AN. Let G be the rotation of point A by angle 7pi/12 around point X. Point C is the orthocenter of triangle GXN. Point H is the centroid of triangle GXN. Find the circumradius of triangle NHC. Express your answer as a decimal number rounded to 2 decimal p... | 0.84 | MATH |
[
{
"content": "Let VM = 5. Let L be the midpoint of segment VM. Let I be the rotation of point M by angle 5pi/6 around point L. Let D be the circumcenter of triangle LIV. Line G passes through point M parallel to segment IV. Point E is the reflection of point I across line G. Let Y be the midpoint of segment VD.... | 9.31 | MATH |
[
{
"content": "AI = 5. Let J be the rotation of point I by angle 7pi/12 around point A. Point R is the reflection of point J across segment IA. Point X is the orthocenter of triangle RAJ. Let E be the midpoint of segment AJ. What is the measure of angle EAX, in radians? Express your answer as a decimal number ro... | 1.83 | MATH |
[
{
"content": "Let KH = 10. Point P is the reflection of point K across point H. Line O is the perpendicular bisector of segment PK. Let C be the rotation of point P by angle pi/12 around point K. Let G be the midpoint of segment CK. Let J be the midpoint of segment HC. Point U is the midpoint of segment CG. Let... | 16.41 | MATH |
[
{
"content": "Let XV = 2. Point Y is obtained by rotating point V by angle pi/2 about point X. Line A is the angle bisector of angle YVX. Point M is the reflection of point X across line A. Let G be the midpoint of segment VX. Point Z is the reflection of point M across segment XY. Find the distance between poi... | 2.12 | MATH |
[
{
"content": "Let UY = 10. Point N is obtained by rotating point U by angle pi/3 about point Y. Let A be the reflection of point Y across point U. Circle Q is the incircle of triangle ANY. Point S is the inversion of point A with respect to circle Q. What is the length of segment NS? Express your answer as a de... | 5.71 | MATH |
[
{
"content": "JV = 6. Point F is obtained by rotating point V by angle pi/6 about point J. Point S is the midpoint of segment FJ. Let N be the line through point J perpendicular to segment JF. Let Y be the line through point F parallel to line N. Point X is the reflection of point J across line Y. Let H be the ... | 9.32 | MATH |
[
{
"content": "EZ = 11. Let D be the reflection of point E across point Z. Let B be the rotation of point D by angle 2pi/3 around point E. Let R be the perpendicular bisector of segment BZ. Point N is the reflection of point D across line R. Let H be the centroid of triangle EBN. What is the distance between poi... | 4.16 | MATH |
[
{
"content": "NH = 1. Let M be the reflection of point H across point N. Point E is obtained by rotating point N by angle pi/6 about point M. Point S is obtained by rotating point E by angle 11pi/12 about point N. Point L is the reflection of point N across point E. Let F be the perpendicular bisector of segmen... | 0.53 | MATH |
[
{
"content": "ED = 12. Let O be the midpoint of segment ED. Let B be the rotation of point O by angle pi/4 around point E. Let F be the reflection of point D across point B. Point I is the reflection of point F across point O. Let J be the centroid of triangle DEB. Find the circumradius of triangle BJI. Express... | 21.69 | MATH |
[
{
"content": "OK = 8. Line B is the perpendicular bisector of segment OK. Let T be the midpoint of segment OK. Line W is the reflection of line B across point O. Point P is on line W with |PK| = 4. Point I is the orthocenter of triangle TPO. Point Q is the reflection of point I across line B. Let N be the incir... | 1.95 | MATH |
[
{
"content": "Let LE = 1. Let N be the rotation of point E by angle pi/12 around point L. Point T is the reflection of point E across point N. Let W be the circumcenter of triangle NLE. Point Z is the reflection of point W across segment NL. Line I passes through points Z and W. Segment LE and line I intersect ... | 0.58 | MATH |
[
{
"content": "Let DK = 10. Let I be the reflection of point K across point D. Line B passes through point I perpendicular to segment KD. Let V be the rotation of point K by angle 7pi/12 around point D. Point M is the reflection of point D across line B. Let F be the line through point V parallel to segment KD. ... | 188.39 | MATH |
[
{
"content": "DF = 8. Point H is obtained by rotating point F by angle 3pi/4 about point D. Let M be the reflection of point H across point D. Point N is the midpoint of segment DH. Point X is the reflection of point H across segment FD. Point B is the reflection of point X across point M. Point I is the reflec... | 79.36 | MATH |
[
{
"content": "ZW = 1. Let M be the rotation of point W by angle 2pi/3 around point Z. Point S is the orthocenter of triangle ZWM. Point F is obtained by rotating point S by angle pi/6 about point Z. Let J be the rotation of point F by angle 7pi/12 around point W. Line T is the perpendicular bisector of segment ... | 0.86 | MATH |
[
{
"content": "Let GA = 6. Point P is the midpoint of segment GA. Point U is obtained by rotating point P by angle pi/4 about point G. Let J be the midpoint of segment UA. Let T be the midpoint of segment PJ. Point L is the reflection of point U across point P. What is the inradius of triangle LJT? Express your ... | 0.24 | MATH |
[
{
"content": "Let SE = 8. Point V is obtained by rotating point E by angle 5pi/6 about point S. Let G be the rotation of point V by angle 5pi/12 around point E. Let Y be the midpoint of segment SG. Point U is the reflection of point E across point S. Let W be the orthocenter of triangle UGY. What is the distanc... | 17.85 | MATH |
[
{
"content": "Let VH = 8. Let Y be the rotation of point V by angle pi/4 around point H. Let C be the reflection of point Y across point V. Point B is obtained by rotating point Y by angle 11pi/12 about point C. Point K is the reflection of point V across point B. Let W be the midpoint of segment HV. Point N is... | 26.16 | MATH |
[
{
"content": "Let HJ = 6. Point I is obtained by rotating point H by angle 5pi/12 about point J. Let K be the orthocenter of triangle IHJ. Point Z is the reflection of point J across segment IH. Point X is the reflection of point Z across segment HJ. What is the distance between points K and X? Express your ans... | 9.21 | MATH |
[
{
"content": "Let QU = 1. Point R is obtained by rotating point Q by angle 5pi/6 about point U. Point D is the reflection of point R across point Q. Point I is the midpoint of segment RQ. Point W is the midpoint of segment UD. What is the circumradius of triangle WIU? Express your answer as a decimal number rou... | 0.73 | MATH |
[
{
"content": "MF = 9. Let J be the vector from point F to point M. Let A be point M translated by vector J. Let X be the rotation of point M by angle 5pi/6 around point A. Line T is the perpendicular bisector of segment AF. Point N is the midpoint of segment AX. Point G is the reflection of point X across line ... | 22.59 | MATH |
[
{
"content": "Let UD = 12. Let K be the midpoint of segment DU. Let B be the rotation of point D by angle pi/3 around point K. Point J is the midpoint of segment DK. Let N be the rotation of point B by angle 7pi/12 around point D. Let L be the rotation of point D by angle 11pi/12 around point J. Let R be the pe... | 12.15 | MATH |
[
{
"content": "Let KY = 2. Let Q be the midpoint of segment KY. Point C is obtained by rotating point Q by angle 3pi/4 about point Y. Let L be the midpoint of segment YC. Let N be the orthocenter of triangle YCK. Let G be the circumcenter of triangle YCK. Let H be the line through points K and L. Point W is the ... | 1.27 | MATH |
[
{
"content": "XP = 11. Let L be the vector from point X to point P. Let J be point P translated by vector L. Let U be the rotation of point J by angle 3pi/4 around point P. Let I be the reflection of point X across point U. Point A is the midpoint of segment IU. Let S be the reflection of point A across point J... | 42.57 | MATH |
[
{
"content": "BU = 2. Point M is obtained by rotating point U by angle pi/6 about point B. Let P be the rotation of point B by angle 2pi/3 around point M. Let S be the perpendicular bisector of segment UB. Let O be the rotation of point U by angle pi/4 around point P. Line G passes through point O parallel to l... | 0.8 | MATH |
[
{
"content": "Let DH = 7. Let X be the rotation of point H by angle 7pi/12 around point D. Point W is the midpoint of segment HD. Point C is the midpoint of segment WH. Point R is the circumcenter of triangle XCD. What is the measure of angle RXH, in radians? Express your answer as a decimal number rounded to 2... | 9.86 | MATH |
[
{
"content": "CD = 2. Point W is the reflection of point C across point D. Point N is obtained by rotating point W by angle pi/4 about point D. Let H be the centroid of triangle DCN. Line S is the perpendicular bisector of segment HC. Point Q is the reflection of point D across line S. Point U is the reflection... | 3.19 | MATH |
[
{
"content": "CZ = 2. Let O be the perpendicular bisector of segment ZC. Let W be the midpoint of segment CZ. Point M is obtained by rotating point Z by angle pi/4 about point C. Let X be the rotation of point C by angle pi/4 around point W. Point Y is obtained by rotating point X by angle pi/3 about point Z. P... | 0.48 | MATH |
[
{
"content": "ZN = 11. Let I be the rotation of point Z by angle pi/12 around point N. Point Y is the reflection of point N across segment ZI. Let V be the reflection of point Y across point Z. Line A passes through point V perpendicular to segment ZI. Line S passes through point V perpendicular to line A. Poin... | 31.32 | MATH |
[
{
"content": "Let AK = 3. Line I is the perpendicular bisector of segment KA. Let G be the rotation of point A by angle 11pi/12 around point K. Let V be the reflection of point G across point K. Point W is the reflection of point G across line I. Let P be the circumcircle of triangle AVW. Point T is the inversi... | 0.26 | MATH |
[
{
"content": "HD = 6. Let Z be the rotation of point H by angle 2pi/3 around point D. Let S be the orthocenter of triangle DZH. Let J be the reflection of point Z across point S. Let T be the midpoint of segment DJ. Find the length of segment ZT. Express your answer as a decimal number rounded to 2 decimal plac... | 13.08 | MATH |
[
{
"content": "Let TW = 6. Point P is obtained by rotating point W by angle pi/4 about point T. Point I is the midpoint of segment PT. Point C is the midpoint of segment TW. Point A is the reflection of point P across segment TW. What is the circumradius of triangle ICA? Express your answer as a decimal number r... | 31.18 | MATH |
[
{
"content": "LI = 3. Point W is obtained by rotating point I by angle 5pi/6 about point L. Let X be the reflection of point W across point I. Point Y is the midpoint of segment XI. Point E is the reflection of point Y across point W. Find the circumradius of triangle ELW. Express your answer as a decimal numbe... | 22.44 | MATH |
[
{
"content": "RQ = 5. Let J be the rotation of point R by angle pi/2 around point Q. Let D be the centroid of triangle RJQ. Point E is the reflection of point D across segment RJ. Let L be the midpoint of segment RQ. What is the distance between points L and E? Express your answer as a decimal number rounded to... | 3.44 | MATH |
[
{
"content": "Let CP = 5. Point U is obtained by rotating point C by angle pi/4 about point P. Point V is the circumcenter of triangle PUC. Point H is the orthocenter of triangle PUC. Point D is the midpoint of segment HC. What is the distance between points V and D? Express your answer as a decimal number roun... | 1.49 | MATH |
[
{
"content": "Let CW = 6. Let Y be the rotation of point W by angle pi/12 around point C. Let A be the circumcenter of triangle CWY. Let Z be the reflection of point C across point W. Let Q be the reflection of point A across point Z. What is the distance between points Y and Q? Express your answer as a decimal... | 14.42 | MATH |
[
{
"content": "Let AB = 7. Point Y is obtained by rotating point B by angle pi/2 about point A. Let L be the reflection of point A across point B. Let D be the midpoint of segment YB. Let C be the midpoint of segment LB. What is the length of segment YC? Express your answer as a decimal number rounded to 2 decim... | 12.62 | MATH |
[
{
"content": "CQ = 2. Let N be the midpoint of segment CQ. Let O be the vector from point Q to point N. Let Z be point C translated by vector O. Point K is obtained by rotating point Q by angle 11pi/12 about point Z. Point F is the reflection of point Q across segment NK. Point W is the reflection of point K ac... | 1.59 | MATH |
[
{
"content": "CB = 3. Point Q is the midpoint of segment CB. Point F is the midpoint of segment QB. Let P be the rotation of point Q by angle pi/6 around point F. Point D is the orthocenter of triangle CPQ. Point X is the reflection of point P across point D. Let S be the midpoint of segment XF. Let U be the mi... | 0.03 | MATH |
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