Datasets:

TTTXXX01
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omega-combined-split-geometry_circle

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train · 1k rows

id stringlengths 26 26	setting_key stringclasses 1 value	setting_name stringclasses 0 values	messages listlengths 1 1	ground_truth stringlengths 3 7	dataset stringclasses 1 value
geometry_circle_train_0200	geometry_circle	null	[ { "role": "user", "content": "HI = 3. Let E be the rotation of point H by angle pi/3 around point I. Circle Q passes through points I, E, and H. Let G be the reflection of point H across point E. Let M be the tangent line from point I to circle Q. Point J is the reflection of point I across point H. Point D...	10.82	math
geometry_circle_train_0201	geometry_circle	null	[ { "role": "user", "content": "Let Z be the circle with center Y and radius 2. P has center B and radius 10 and is externally tangent to circle Z. Let D be the rotation of point Y by angle pi/6 around point B. Let R be the rotation of point Y by angle 11pi/12 around point D. Point M is the midpoint of segmen...	63.91	math
geometry_circle_train_0202	geometry_circle	null	[ { "role": "user", "content": "Let G be the circle with center J and radius 3. U has center B and radius 9 and is externally tangent to circle G. Point S is the reflection of point B across point J. Let Z be the polar line of point S relative to circle U. Let D be the reflection of point J across point S. Po...	29.25	math
geometry_circle_train_0203	geometry_circle	null	[ { "role": "user", "content": "Let T be the circle with center F and radius 6. B has center Z and radius 3 and is internally tangent to circle T. Let D be the reflection of point F across point Z. Point Y is the midpoint of segment FZ. Line Q is the polar of point D with respect to circle B. Point K is the r...	6.3	math
geometry_circle_train_0204	geometry_circle	null	[ { "role": "user", "content": "Let Z be the circle with center N and radius 7. Circle G has center N and radius 6. Chord MW of circle G has length 5. NB = 1. Point V is obtained by rotating point W by angle pi/2 about point B. Point K is the midpoint of segment MW. Point J is obtained by rotating point B by ...	2.5	math
geometry_circle_train_0205	geometry_circle	null	[ { "role": "user", "content": "Circle L has center V and radius 5. Chord AQ of circle L has length 1. Let H be the perpendicular bisector of segment AQ. Circle U has center V and passes through point A. Let M be the circle with center Q and radius 1. Point N is obtained by rotating point V by angle 7pi/12 ab...	7.61	math
geometry_circle_train_0206	geometry_circle	null	[ { "role": "user", "content": "HY = 1. Let V be the rotation of point Y by angle 5pi/12 around point H. Let C be the reflection of point V across point Y. Line L is the angle bisector of angle VCH. Line D passes through point H perpendicular to segment HY. Point Q is the reflection of point V across line L. ...	3.79	math
geometry_circle_train_0207	geometry_circle	null	[ { "role": "user", "content": "VR = 12. Let M be the rotation of point R by angle 11pi/12 around point V. Let C be the circle with center R passing through point V. Let I be the rotation of point R by angle 5pi/6 around point V. Line S is the angle bisector of angle MRV. Point B is the reflection of point V ...	21.45	math
geometry_circle_train_0208	geometry_circle	null	[ { "role": "user", "content": "Circle W has center J and radius 3. Point R is on circle W. Point Q is obtained by rotating point J by angle pi about point R. Let P be the inversion of point Q in circle W. Let Z be the rotation of point P by angle pi/12 around point J. Point O is the midpoint of segment RZ. L...	6.62	math
geometry_circle_train_0209	geometry_circle	null	[ { "role": "user", "content": "Let LV = 9. Let Y be the circle with center V passing through point L. Line A is the tangent from point L to circle Y. Circle Q has center L and radius 5. Let H be the vector from point V to point L. Point U is the midpoint of segment LV. Circle S is the reflection of circle Q ...	15.98	math
geometry_circle_train_0210	geometry_circle	null	[ { "role": "user", "content": "Circle E has center O and radius 7. Chord LN of circle E has length 3. Point J is the midpoint of segment LN. Let D be the rotation of point L by angle 7pi/12 around point J. Circle K has center O and passes through point D. Point Z is obtained by rotating point O by angle pi/4...	9.25	math
geometry_circle_train_0211	geometry_circle	null	[ { "role": "user", "content": "Let W be the circle with center H and radius 2. Let O be the circle with center H and radius 10. Point T is on circle W. Circle L has center H and radius 6. Let J be the reflection of point T across point H. Let Y be the reflection of point H across point J. Let U be the invers...	1.22	math
geometry_circle_train_0212	geometry_circle	null	[ { "role": "user", "content": "Let M be the circle with center P and radius 9. Point Y lies on circle M. Line H is the perpendicular bisector of segment YP. Circle A has center P and radius 3. Let N be the vector from point P to point Y. Let J be the reflection of circle A across line H. Let D be the inversi...	16.04	math
geometry_circle_train_0213	geometry_circle	null	[ { "role": "user", "content": "Let F be the circle with center H and radius 9. M has center B and radius 11 and is externally tangent to circle F. Let T be the midpoint of segment HB. Line P is the polar of point T with respect to circle F. Let S be the inversion of point B in circle F. Let U be the line thr...	3.54	math
geometry_circle_train_0214	geometry_circle	null	[ { "role": "user", "content": "Let RB = 2. Let C be the perpendicular bisector of segment BR. Circle P has center R and passes through point B. Point K is obtained by rotating point R by angle 11pi/12 about point B. Line G is the reflection of line C across point B. Point H is the reflection of point B acros...	1.81	math
geometry_circle_train_0215	geometry_circle	null	[ { "role": "user", "content": "Let AC = 9. Point K is obtained by rotating point A by angle pi/6 about point C. Let V be the midpoint of segment CK. Let F be the midpoint of segment VC. Let E be the circle with center V and radius 1. Line L is the perpendicular bisector of segment VC. Line I is the polar of ...	4.75	math
geometry_circle_train_0216	geometry_circle	null	[ { "role": "user", "content": "Circle J has center P and radius 6. Chord HI of circle J has length 4. Let V be the rotation of point I by angle pi around point H. Point N is the reflection of point P across point H. Let Z be the inversion of point N in circle J. Line W passes through point Z perpendicular to...	6.53	math
geometry_circle_train_0217	geometry_circle	null	[ { "role": "user", "content": "Let S be the circle with center U and radius 1. Point R is on circle S. Let I be the perpendicular bisector of segment RU. Let L be the midpoint of segment RU. Let J be the inversion of point L in circle S. Point T lies on line I at distance 10 from point R. Let V be the circle...	5.86	math
geometry_circle_train_0218	geometry_circle	null	[ { "role": "user", "content": "Let P be the circle with center D and radius 2. W has center K and radius 1 and is externally tangent to circle P. Circle Q has center K and radius 10. Let Z be the rotation of point K by angle 3pi/4 around point D. Let C be the inversion of point Z in circle P. Let A be the re...	10.69	math
geometry_circle_train_0219	geometry_circle	null	[ { "role": "user", "content": "BW = 3. Point M is obtained by rotating point B by angle pi/12 about point W. Let H be the rotation of point W by angle 2pi/3 around point M. Let P be the midpoint of segment HW. Line A is the angle bisector of angle WHB. Circle N has center P and radius 2. Line O passes throug...	3.9	math
geometry_circle_train_0220	geometry_circle	null	[ { "role": "user", "content": "RA = 9. Line E is the perpendicular bisector of segment AR. Circle X has center A and radius 9. Point J is the midpoint of segment AR. Point H is on line E with \|HJ\| = 1. Point K is the inversion of point H with respect to circle X. Let U be the reflection of point K across poi...	51.6	math
geometry_circle_train_0221	geometry_circle	null	[ { "role": "user", "content": "Let CR = 1. Let N be the rotation of point R by angle pi/3 around point C. Let Q be the reflection of point N across point C. Let M be the rotation of point R by angle pi/12 around point C. Let S be the reflection of point R across point M. Let Y be the circle with center R pas...	0.32	math
geometry_circle_train_0222	geometry_circle	null	[ { "role": "user", "content": "Let CK = 10. Line Y passes through point C perpendicular to segment KC. Point R lies on line Y at distance 4 from point K. Let I be the vector from point C to point K. Line D passes through points R and K. Circle L passes through points C, R, and K. Point F is the translation o...	22.16	math
geometry_circle_train_0223	geometry_circle	null	[ { "role": "user", "content": "Let UM = 10. Let V be the reflection of point U across point M. Let T be the rotation of point U by angle 5pi/6 around point M. Line P passes through points U and T. Let J be the midpoint of segment TU. Let N be the circle with center J and radius 8. Point I is the reflection o...	208.05	math
geometry_circle_train_0224	geometry_circle	null	[ { "role": "user", "content": "Let R be the circle with center U and radius 4. Circle Z has center U and radius 10. Point O lies on circle Z. Let M be the rotation of point O by angle 5pi/12 around point U. Point T is the reflection of point U across point O. Let F be the reflection of point M across point U...	16.32	math
geometry_circle_train_0225	geometry_circle	null	[ { "role": "user", "content": "Circle F has center M and radius 9. X has center D and radius 5 and is externally tangent to circle F. Let J be the reflection of point D across point M. Point I is the inversion of point J with respect to circle X. Point V is obtained by rotating point M by angle pi about poin...	16.59	math
geometry_circle_train_0226	geometry_circle	null	[ { "role": "user", "content": "Circle T has center H and radius 10. Point C lies on circle T. Line D is the polar of point C with respect to circle T. Point E is on line D with \|EC\| = 2. Point Y is the reflection of point H across line D. Point U is on line D with \|UY\| = 10. Let W be the reflection of point ...	14.14	math
geometry_circle_train_0227	geometry_circle	null	[ { "role": "user", "content": "Let E be the circle with center M and radius 10. D has center K and radius 2 and is internally tangent to circle E. Let B be the reflection of point M across point K. Let V be the reflection of circle D through point B. Line C passes through points K and M. Line P is the polar ...	15.83	math
geometry_circle_train_0228	geometry_circle	null	[ { "role": "user", "content": "Let OZ = 11. Let A be the rotation of point O by angle pi/4 around point Z. Let Q be the circle through points O, A, and Z. Point N is the reflection of point Z across segment AO. Point C is the center of circle Q. Point B is the midpoint of segment NO. Point J is the midpoint ...	12.45	math
geometry_circle_train_0229	geometry_circle	null	[ { "role": "user", "content": "Let P be the circle with center L and radius 9. Chord DT of circle P has length 9. Point W is obtained by rotating point D by angle 3pi/4 about point L. Let Y be the line through point T perpendicular to segment DT. Point C is the reflection of point D across point T. Point N i...	19.35	math
geometry_circle_train_0230	geometry_circle	null	[ { "role": "user", "content": "Circle C has center O and radius 11. Let OR = 2. Let U be the reflection of circle C through point R. Let J be the circle with center O and radius 3. Point I is the inversion of point R with respect to circle U. Let D be the midpoint of segment IO. Let E be the rotation of poin...	58.27	math
geometry_circle_train_0231	geometry_circle	null	[ { "role": "user", "content": "YS = 8. Circle J has center S and radius 12. Let D be the reflection of point S across point Y. Chord IK of circle J has length 1. Circle M passes through points K, I, and Y. Point N is the center of circle M. Let C be the midpoint of segment IK. Point R is the reflection of po...	5.99	math
geometry_circle_train_0232	geometry_circle	null	[ { "role": "user", "content": "Let R be the circle with center O and radius 10. A has center L and radius 3 and is internally tangent to circle R. Point P is obtained by rotating point L by angle pi/4 about point O. Point V is obtained by rotating point P by angle 7pi/12 about point L. Let N be the angle bis...	27.48	math
geometry_circle_train_0233	geometry_circle	null	[ { "role": "user", "content": "Let O be the circle with center K and radius 5. S has center E and radius 4 and is externally tangent to circle O. Let C be the circle with center K and radius 8. P has center Z and radius 11 and is externally tangent to circles O and S. Point U is the midpoint of segment EZ. L...	1.89	math
geometry_circle_train_0234	geometry_circle	null	[ { "role": "user", "content": "Let P be the circle with center M and radius 10. Chord KI of circle P has length 3. Let L be the circle with center K and radius 6. Circle G is the reflection of circle L across point M. Line N is the polar of point I with respect to circle G. Line J is the perpendicular bisect...	192.53	math
geometry_circle_train_0235	geometry_circle	null	[ { "role": "user", "content": "Let TE = 5. Let B be the circle with center T passing through point E. Point D is the reflection of point E across point T. Point U is the reflection of point E across point D. Let G be the reflection of point T across point E. Let N be the inversion of point G in circle B. Wha...	17.5	math
geometry_circle_train_0236	geometry_circle	null	[ { "role": "user", "content": "Let T be the circle with center S and radius 6. Chord FQ of circle T has length 5. Let J be the line through point F perpendicular to segment FQ. Line C passes through point Q parallel to line J. Point A is the reflection of point F across line C. Let L be the midpoint of segme...	6.48	math
geometry_circle_train_0237	geometry_circle	null	[ { "role": "user", "content": "Let OG = 8. Point F is obtained by rotating point O by angle 3pi/4 about point G. Circle T passes through points F, O, and G. Point N is the center of circle T. Circle R is the reflection of circle T across point F. Let J be the midpoint of segment ON. Point X is the inversion ...	16.03	math
geometry_circle_train_0238	geometry_circle	null	[ { "role": "user", "content": "Let HE = 5. Let A be the reflection of point H across point E. Let R be the midpoint of segment EA. Let U be the circle with center R passing through point H. Let O be the reflection of point R across point E. Let T be the inversion of point O in circle U. Find the distance bet...	3.75	math
geometry_circle_train_0239	geometry_circle	null	[ { "role": "user", "content": "RJ = 5. Let H be the perpendicular bisector of segment JR. Let D be the reflection of point J across point R. Point F is obtained by rotating point J by angle 3pi/4 about point D. Point B is the reflection of point F across line H. Circle G passes through points J, B, and R. Ci...	31.07	math
geometry_circle_train_0240	geometry_circle	null	[ { "role": "user", "content": "Circle A has center F and radius 4. Point H is on circle A. Let I be the rotation of point H by angle 7pi/12 around point F. Let Q be the line through points F and H. Line G is the reflection of line Q across point I. Circle E is the reflection of circle A across line G. Let N ...	7.98	math
geometry_circle_train_0241	geometry_circle	null	[ { "role": "user", "content": "Let IC = 1. Let Q be the rotation of point I by angle 2pi/3 around point C. Let D be the circle with center C and radius 3. Line T passes through points Q and I. Point L is the inversion of point I with respect to circle D. Let G be the reflection of circle D through point I. L...	0.52	math
geometry_circle_train_0242	geometry_circle	null	[ { "role": "user", "content": "NK = 3. Let S be the rotation of point K by angle 5pi/12 around point N. Let Q be the circle through points K, S, and N. Point J is the midpoint of segment NS. Let G be the center of circle Q. Circle O passes through points G, J, and N. What is the radius of circle O? Express y...	0.95	math
geometry_circle_train_0243	geometry_circle	null	[ { "role": "user", "content": "GQ = 12. Line C is the perpendicular bisector of segment GQ. Point V is on line C with \|VG\| = 3. Let U be the circle through points V, G, and Q. Let N be the polar line of point G relative to circle U. Point O is the reflection of point V across line N. Circle R passes through ...	6.43	math
geometry_circle_train_0244	geometry_circle	null	[ { "role": "user", "content": "MJ = 1. Let E be the midpoint of segment MJ. Let P be the circle with center E and radius 3. Circle R is the reflection of circle P across point J. Let A be the circle with center J and radius 7. Let S be the rotation of point E by angle pi/2 around point M. Point D is the inve...	2.73	math
geometry_circle_train_0245	geometry_circle	null	[ { "role": "user", "content": "MQ = 12. Let E be the line through points M and Q. Line C is the perpendicular bisector of segment QM. Let R be the reflection of line C through point Q. Let Y be the circle with center Q passing through point M. Line W is the reflection of line R across line C. Point K is obta...	49.94	math
geometry_circle_train_0246	geometry_circle	null	[ { "role": "user", "content": "Let V be the circle with center P and radius 7. S has center U and radius 2 and is externally tangent to circle V. Line O is the polar of point P with respect to circle S. Point I lies on line O at distance 7 from point P. Point J is the inversion of point I with respect to cir...	11.35	math
geometry_circle_train_0247	geometry_circle	null	[ { "role": "user", "content": "Circle T has center C and radius 6. Let U be the circle with center C and radius 7. H has center B and radius 5 and is internally tangent to circle T. Let N be the polar line of point B relative to circle U. Line Y is the reflection of line N across point C. Let F be the rotati...	101.0	math
geometry_circle_train_0248	geometry_circle	null	[ { "role": "user", "content": "Let E be the circle with center M and radius 7. Chord NH of circle E has length 5. Circle O has center M and radius 11. Point P is the reflection of point H across point M. Let K be the polar line of point P relative to circle O. Point S is on line K. Point Q is the reflection ...	2.02	math
geometry_circle_train_0249	geometry_circle	null	[ { "role": "user", "content": "LC = 5. Point X is obtained by rotating point C by angle pi/6 about point L. Point O is the reflection of point L across point X. Let P be the reflection of point O across point C. Let J be the rotation of point O by angle pi/12 around point X. Let Q be the midpoint of segment ...	2.54	math
geometry_circle_train_0250	geometry_circle	null	[ { "role": "user", "content": "LS = 9. Let N be the midpoint of segment SL. Let Q be the rotation of point N by angle pi/12 around point L. Let G be the line through points Q and L. Let V be the reflection of point L across point Q. Let X be the reflection of point N across point L. Let D be the line through...	18.39	math
geometry_circle_train_0251	geometry_circle	null	[ { "role": "user", "content": "ME = 2. Point O is the midpoint of segment EM. Let P be the rotation of point O by angle pi around point M. Line H is the angle bisector of angle PMO. Circle L has center M and passes through point E. Let W be the midpoint of segment OE. Let U be the inversion of point O in cir...	8.47	math
geometry_circle_train_0252	geometry_circle	null	[ { "role": "user", "content": "DN = 4. Line J passes through points D and N. Point F is the midpoint of segment ND. Let X be the midpoint of segment DF. Point H is obtained by rotating point D by angle 3pi/4 about point X. Line I is the reflection of line J across point H. Let V be the midpoint of segment DH...	4.25	math
geometry_circle_train_0253	geometry_circle	null	[ { "role": "user", "content": "Let ZS = 6. Point M is the reflection of point Z across point S. Let F be the circle with center Z passing through point S. Let L be the inversion of point M in circle F. Point K is the midpoint of segment LZ. What is the distance between points K and L? Express your answer as ...	1.5	math
geometry_circle_train_0254	geometry_circle	null	[ { "role": "user", "content": "Let SN = 2. Let R be the reflection of point S across point N. Point M is obtained by rotating point S by angle 2pi/3 about point N. Point G is the midpoint of segment NR. Let Z be the circle with center M and radius 8. Point I is obtained by rotating point M by angle 11pi/12 a...	0.92	math
geometry_circle_train_0255	geometry_circle	null	[ { "role": "user", "content": "Let Z be the circle with center C and radius 11. CW = 1. Line U passes through points W and C. Let F be the polar line of point W relative to circle Z. Line A is the reflection of line F across point C. Let R be the midpoint of segment WC. Point K is on line U. Point H lies on ...	120.75	math
geometry_circle_train_0256	geometry_circle	null	[ { "role": "user", "content": "Circle M has center N and radius 6. Chord OU of circle M has length 11. Point C is obtained by rotating point O by angle pi/12 about point U. Let X be the midpoint of segment OU. Let Z be the rotation of point N by angle pi/2 around point C. Circle R passes through points Z, O,...	131.71	math
geometry_circle_train_0257	geometry_circle	null	[ { "role": "user", "content": "Circle B has center S and radius 8. A has center Z and radius 11 and is externally tangent to circle B. Let N be the rotation of point Z by angle pi/2 around point S. Let W be the circle through points Z, N, and S. Let M be the inversion of point S in circle A. Let V be the mid...	11.41	math
geometry_circle_train_0258	geometry_circle	null	[ { "role": "user", "content": "Let C be the circle with center Y and radius 2. S has center O and radius 2 and is externally tangent to circle C. Let E be the perpendicular bisector of segment OY. Point D is the inversion of point Y with respect to circle S. Line G is the perpendicular bisector of segment DO...	2.47	math
geometry_circle_train_0259	geometry_circle	null	[ { "role": "user", "content": "Let J be the circle with center X and radius 2. P has center O and radius 3 and is externally tangent to circle J. Point H is the reflection of point X across point O. Line C is the polar of point X with respect to circle P. Point E is the inversion of point H with respect to c...	10.4	math
geometry_circle_train_0260	geometry_circle	null	[ { "role": "user", "content": "Let YW = 4. Point A is obtained by rotating point Y by angle 5pi/12 about point W. Let F be the circle with center Y and radius 2. Let V be the circle through points W, A, and Y. Let I be the polar line of point A relative to circle V. Point G lies on line I at distance 6 from ...	2.52	math
geometry_circle_train_0261	geometry_circle	null	[ { "role": "user", "content": "Circle G has center E and radius 8. Chord PD of circle G has length 1. Let N be the rotation of point E by angle pi/3 around point D. Line C is the perpendicular bisector of segment PD. Let F be the midpoint of segment NE. Let L be the reflection of line C through point P. Segm...	1.73	math
geometry_circle_train_0262	geometry_circle	null	[ { "role": "user", "content": "YM = 12. Point I is the reflection of point Y across point M. Let N be the vector from point I to point Y. Let T be the rotation of point Y by angle 3pi/4 around point I. Let O be the reflection of point T across point Y. Let D be the vector from point M to point Y. Point F is ...	23.1	math
geometry_circle_train_0263	geometry_circle	null	[ { "role": "user", "content": "Circle K has center I and radius 12. Point G is on circle K. Let IT = 6. Point H is the midpoint of segment GT. Let Y be the midpoint of segment IG. Find the distance between points H and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the ans...	3.0	math
geometry_circle_train_0264	geometry_circle	null	[ { "role": "user", "content": "Circle W has center I and radius 1. Point B lies on circle W. Let Y be the midpoint of segment BI. Line A is the perpendicular bisector of segment BI. Line E is the polar of point B with respect to circle W. Point L is on line E with \|LY\| = 8. Let O be the reflection of circle ...	10.97	math
geometry_circle_train_0265	geometry_circle	null	[ { "role": "user", "content": "Let S be the circle with center F and radius 6. B has center I and radius 1 and is internally tangent to circle S. Let O be the perpendicular bisector of segment FI. Point M is obtained by rotating point F by angle pi/6 about point I. Let W be the midpoint of segment IF. Line H...	15.24	math
geometry_circle_train_0266	geometry_circle	null	[ { "role": "user", "content": "LS = 1. Let K be the circle with center L and radius 12. Circle Y has center L and passes through point S. Let Z be the midpoint of segment SL. B is constructed so that ZB = 6. Let W be the inversion of point B in circle Y. Point O is the inversion of point W with respect to ci...	0.5	math
geometry_circle_train_0267	geometry_circle	null	[ { "role": "user", "content": "Circle M has center D and radius 1. Let DW = 6. Point Y is obtained by rotating point D by angle 5pi/6 about point W. Point R is the midpoint of segment WY. Point Q is the inversion of point R with respect to circle M. Line Z is the polar of point R with respect to circle M. Le...	14.34	math
geometry_circle_train_0268	geometry_circle	null	[ { "role": "user", "content": "XJ = 7. Point S is obtained by rotating point J by angle pi/12 about point X. Let Z be the rotation of point S by angle pi/3 around point J. Point W is the reflection of point Z across point J. Line K is the perpendicular bisector of segment ZS. Let O be the line through point ...	12.9	math
geometry_circle_train_0269	geometry_circle	null	[ { "role": "user", "content": "Circle O has center F and radius 4. K has center R and radius 2 and is externally tangent to circle O. Let Y be the circle with center R and radius 7. A has center V and radius 8 and is externally tangent to circle Y. Let T be the rotation of point R by angle pi/6 around point ...	25.03	math
geometry_circle_train_0270	geometry_circle	null	[ { "role": "user", "content": "Let IJ = 2. Point E is the reflection of point I across point J. Let M be the rotation of point J by angle pi/6 around point E. Circle T has center E and radius 1. Let Y be the line through point M perpendicular to segment IM. Point O is the inversion of point J with respect to...	9.69	math
geometry_circle_train_0271	geometry_circle	null	[ { "role": "user", "content": "Let U be the circle with center R and radius 8. I has center Z and radius 3 and is internally tangent to circle U. Let M be the circle with center R passing through point Z. Circle F is the reflection of circle I across point R. Let O be the reflection of point R across point Z...	1.82	math
geometry_circle_train_0272	geometry_circle	null	[ { "role": "user", "content": "Circle L has center J and radius 9. Chord BI of circle L has length 1. Point U is obtained by rotating point B by angle pi/3 about point I. Point A is the midpoint of segment BI. Point E is the midpoint of segment JB. Let X be the midpoint of segment BE. Let F be the reflection...	10.31	math
geometry_circle_train_0273	geometry_circle	null	[ { "role": "user", "content": "Let CA = 12. Point J is the reflection of point A across point C. Point H is the midpoint of segment CA. Let D be the circle with center H and radius 1. Point I is the reflection of point C across point J. Let V be the line through point I perpendicular to segment CA. Point S i...	15.05	math
geometry_circle_train_0274	geometry_circle	null	[ { "role": "user", "content": "XY = 3. Line E passes through points X and Y. Point W is obtained by rotating point X by angle pi/6 about point Y. Line K is the reflection of line E across point W. Circle P has center X and radius 4. Let F be the line through point Y perpendicular to line E. Let U be the refl...	3.35	math
geometry_circle_train_0275	geometry_circle	null	[ { "role": "user", "content": "FO = 7. Point G is the midpoint of segment FO. Let X be the midpoint of segment GO. Circle Z has center O and radius 2. Let I be the polar line of point G relative to circle Z. Let B be the reflection of circle Z across line I. Let R be the inversion of point G in circle B. Let...	7.33	math
geometry_circle_train_0276	geometry_circle	null	[ { "role": "user", "content": "Circle S has center L and radius 1. O has center G and radius 1 and is externally tangent to circle S. Let H be the midpoint of segment GL. Let D be the midpoint of segment HG. Chord JW of circle O has length 1. Let F be the rotation of point W by angle pi/12 around point L. Le...	1.37	math
geometry_circle_train_0277	geometry_circle	null	[ { "role": "user", "content": "Let S be the circle with center E and radius 1. L has center X and radius 1 and is externally tangent to circle S. Point T is the reflection of point E across point X. Line H is the polar of point E with respect to circle L. Point B lies on line H at distance 8 from point X. Le...	16.28	math
geometry_circle_train_0278	geometry_circle	null	[ { "role": "user", "content": "Let BQ = 2. Line N is the perpendicular bisector of segment QB. Let C be the vector from point B to point Q. Let E be the line through point B parallel to line N. Let G be the midpoint of segment BQ. Point F lies on line E at distance 9 from point B. Let L be point F translated...	11.95	math
geometry_circle_train_0279	geometry_circle	null	[ { "role": "user", "content": "Let IS = 5. Point O is the reflection of point I across point S. Point J is the reflection of point I across point O. Circle H has center J and radius 3. Point W is the inversion of point S with respect to circle H. Let X be the midpoint of segment SI. Point F is the reflection...	54.4	math
geometry_circle_train_0280	geometry_circle	null	[ { "role": "user", "content": "NY = 5. Let J be the rotation of point Y by angle pi/4 around point N. Point G is obtained by rotating point N by angle 5pi/12 about point J. Let W be the rotation of point N by angle pi/12 around point J. Line H passes through points W and Y. Let O be the circle through points...	7.01	math
geometry_circle_train_0281	geometry_circle	null	[ { "role": "user", "content": "Circle O has center F and radius 2. U has center R and radius 8 and is externally tangent to circle O. Let Z be the midpoint of segment RF. Let D be the circle with center Z passing through point R. Point K is obtained by rotating point F by angle 5pi/6 about point Z. Point P i...	5.26	math
geometry_circle_train_0282	geometry_circle	null	[ { "role": "user", "content": "Let N be the circle with center Y and radius 7. Point I is on circle N. Circle Z has center Y and radius 3. Circle O has center I and passes through point Y. Let B be the rotation of point Y by angle 5pi/12 around point I. Let M be the circle through points Y, I, and B. Point E...	4.17	math
geometry_circle_train_0283	geometry_circle	null	[ { "role": "user", "content": "Let I be the circle with center V and radius 3. Chord AM of circle I has length 2. Let C be the reflection of point A across point M. Let B be the line through point C perpendicular to segment AM. Let G be the midpoint of segment AM. Point O is the reflection of point V across ...	8.83	math
geometry_circle_train_0284	geometry_circle	null	[ { "role": "user", "content": "Circle O has center R and radius 3. Let X be the circle with center R and radius 5. Let RF = 7. Let P be the reflection of point R across point F. Let Y be the inversion of point P in circle O. Point S is the reflection of point F across point Y. Let T be the inversion of point...	5.02	math
geometry_circle_train_0285	geometry_circle	null	[ { "role": "user", "content": "UI = 8. Let H be the rotation of point I by angle pi around point U. Point B is the reflection of point H across point I. Let O be the perpendicular bisector of segment UB. Line Q is the reflection of line O across point B. Let G be the rotation of point I by angle pi around po...	33.55	math
geometry_circle_train_0286	geometry_circle	null	[ { "role": "user", "content": "Let B be the circle with center F and radius 4. U has center P and radius 6 and is externally tangent to circle B. Let C be the rotation of point P by angle pi/12 around point F. Let Q be the reflection of circle U through point F. Let G be the circle with center C passing thro...	7.19	math
geometry_circle_train_0287	geometry_circle	null	[ { "role": "user", "content": "Let OQ = 4. Let P be the vector from point O to point Q. Let V be the rotation of point O by angle pi/12 around point Q. Let Z be the rotation of point Q by angle pi/2 around point V. Let J be the circle with center Z and radius 3. Point K is the midpoint of segment ZO. Let B b...	11.65	math
geometry_circle_train_0288	geometry_circle	null	[ { "role": "user", "content": "Let TP = 9. Let M be the midpoint of segment TP. Let L be the perpendicular bisector of segment PM. Let U be the rotation of point T by angle pi/6 around point M. Point Z is the reflection of point T across line L. Point Y is the reflection of point U across segment PM. Let O b...	16.47	math
geometry_circle_train_0289	geometry_circle	null	[ { "role": "user", "content": "MI = 7. Let R be the rotation of point I by angle 7pi/12 around point M. Circle P has center R and radius 3. Line D is the angle bisector of angle RMI. Let Y be the rotation of point I by angle pi around point R. Line W is the perpendicular bisector of segment RM. Line X is the...	14.64	math
geometry_circle_train_0290	geometry_circle	null	[ { "role": "user", "content": "Let LN = 5. LS = 2. Let I be the midpoint of segment NS. Circle U has center I and radius 9. F has center E and radius 3 and is internally tangent to circle U. Point C is the midpoint of segment SE. Let Y be the line through points C and N. Point J is the intersection of line Y...	2.0	math
geometry_circle_train_0291	geometry_circle	null	[ { "role": "user", "content": "Circle H has center R and radius 10. Z has center D and radius 6 and is externally tangent to circle H. Let W be the polar line of point D relative to circle H. Point U is the midpoint of segment DR. Point G is obtained by rotating point U by angle pi about point D. Let F be th...	16.74	math
geometry_circle_train_0292	geometry_circle	null	[ { "role": "user", "content": "Let PE = 6. Let M be the perpendicular bisector of segment EP. Point I is the midpoint of segment PE. Point X lies on line M at distance 11 from point I. Let A be the midpoint of segment IP. Point C is the reflection of point X across segment EP. Point Y is the midpoint of segm...	2.85	math
geometry_circle_train_0293	geometry_circle	null	[ { "role": "user", "content": "Circle Z has center K and radius 2. Point U is on circle Z. Point F is the midpoint of segment KU. M has center P and radius 2 and is externally tangent to circle Z. Let R be the circle with center K passing through point P. Let W be the inversion of point F in circle R. Point ...	8.5	math
geometry_circle_train_0294	geometry_circle	null	[ { "role": "user", "content": "Let U be the circle with center M and radius 10. E has center X and radius 1 and is externally tangent to circle U. Point H is the reflection of point M across point X. Let B be the inversion of point H in circle E. Circle Z is the reflection of circle U across point X. Point Q...	16.47	math
geometry_circle_train_0295	geometry_circle	null	[ { "role": "user", "content": "Circle Q has center E and radius 11. Point X lies on circle Q. Let B be the midpoint of segment EX. Circle I is the reflection of circle Q across point X. Point L is the inversion of point B with respect to circle I. What is the distance between points B and L? Express your ans...	9.17	math
geometry_circle_train_0296	geometry_circle	null	[ { "role": "user", "content": "Let B be the circle with center K and radius 4. Point W lies on circle B. Point N is the reflection of point K across point W. Point C is obtained by rotating point K by angle 2pi/3 about point N. Let Y be the circle with center N passing through point C. Let J be the midpoint ...	16.22	math
geometry_circle_train_0297	geometry_circle	null	[ { "role": "user", "content": "Circle N has center Q and radius 1. Chord PV of circle N has length 2. Line O passes through point V perpendicular to segment PV. Point Y is on line O with \|YV\| = 12. Circle B has center Q and radius 12. Point H is the midpoint of segment YV. Let R be the inversion of point H i...	20.63	math
geometry_circle_train_0298	geometry_circle	null	[ { "role": "user", "content": "KI = 12. Let Y be the circle with center K passing through point I. Circle D is the reflection of circle Y across point I. Let F be the perpendicular bisector of segment KI. Let O be the midpoint of segment KI. Circle G is the reflection of circle D across line F. Point M is th...	12.0	math
geometry_circle_train_0299	geometry_circle	null	[ { "role": "user", "content": "Let ME = 4. Let G be the reflection of point M across point E. Let R be the rotation of point E by angle pi/4 around point G. Let X be the midpoint of segment ME. Let F be the circle through points M, G, and R. Point B is the inversion of point X with respect to circle F. Let Q...	3.82	math

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