Upload prompt.txt

prompt.txt

@@ -57,15 +57,39 @@ Example:
 - simtri a b c p q r : triangles abc and pqr are similar
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 Example Problem Transformation: 
-user: Given a triangle ABC with three acute angles connected to the circle (O). The altitudes AD, BE, CF intersect at H and intersect the circle (O) at M, N, P respectively. Prove that: Quadrilateral CEHD is inscribed
-assistant: a b c = triangle a b c; o = circle o a b c; d = foot d a b c; e = foot e b a c; f = foot f c a b; h = orthocenter h a b c; m = on_line m a d, on_circle m o a; n = on_line n b e, on_circle n o a; p = on_line p c f, on_circle p o a ? cyclic c e h d
+user: Given triangle ABC with three acute angles inscribed in circle (O). The altitudes AD, BE, CF intersect at H and intersect circle (O) at M, N, P respectively. Prove that: Quadrilateral CEHD is inscribed.
+assistant: a b c = triangle a b c; o = circle o a b c; d = foot d a b c; e = foot e b a c; f = foot f c a b; h = orthocenter h a b c; m = on_circle m o a, on_line m a d; n = on_circle n o a, on_line n b e; p = on_circle p o a, on_line p c f ? cyclic c e h d
 
-user: Given a right triangle ABC at A. On side AC, take point M, construct a circle (O) with diameter MC. Line BM intersects circle (O) at D. Line AD intersects circle (O) at S. Prove that ABCD is an inscribed quadrilateral.
-assistant: a b c = r_triangle a b c; m = on_line m a c; o = midpoint o m c; d = on_line d b m, on_circle d o m; s = on_line s a d, on_circle s o m ? cyclic a b c d
+user: Given acute triangle ABC, Draw altitudes AD, BE, CF. Let H be the orthocenter of the triangle. Let M, N, P, Q be the perpendicular projections of D onto AB, BE, CF, AC, respectively. Prove: quadrilateral DPQC is inscribed.
+assistant: a b c = triangle a b c; d = foot d a b c; e = foot e b a c; f = foot f c a b; h = orthocenter h a b c; m = foot m d a b; n = foot n d b e; p = foot p d c f; q = foot q d a c ? cyclic d p q c
 
-user: Given triangle ABC, bisector AD, median AM. The line perpendicular to AD at D intersects AB, AM at X, Y respectively. The line perpendicular to AB at X intersects AD at Z. Prove that YZ is perpendicular to BC.
-assistant: a b c = triangle a b c; d = angle_bisector d b a c, on_line d b c; m = midpoint m b c; x = on_tline x d a d, on_line x a b; y = on_tline y d a d, on_line y a m; z = on_tline z x a b, on_line z a d ? perp y z b c
+user: Given triangle ABC inscribed in a circle with center O. D and E are the midpoints of arcs AB; AC respectively. Call the intersections of DE with AB; AC H and K respectively. Call I the intersection of BE with CD. Prove: CEKI is inscribed.
+assistant: a b c = triangle a b c; o = circle o a b c; d = on_circle d o a, angle_bisector d a o b; e = on_circle e o a, angle_bisector e a o c; h = on_line h d e, on_line h a b; k = on_line k d e, on_line k a c; i = on_line i b e, on_line i c d ? cyclic c e k i
 
-user: Given acute triangle ABC, Draw altitudes AD, BE, CF. Let H be the orthocenter of the triangle. Let M, N, P, Q be the perpendicular projections of D onto AB, BE, CF, AC, respectively. Prove: quadrilateral BMND is inscribed.
-assistant: a b c = triangle a b c; d = foot d a b c; e = foot e b a c; f = foot f c a b; h = orthocenter h a b c; m = foot m d a b; n = foot n d b e; p = foot p d c f; q = foot q d a c ? cyclic b m n d
+user: Given triangle ABC with: ∠A = 90 degrees. D is a point on side AB. The circle with diameter BD intersects BC at E. The lines CD; AE intersect the circle at the second points F and G respectively. Prove that AC // FG.
+assistant: a b c = r_triangle a b c; d = on_line d a b; z = midpoint z b d; e = on_circle e z b, on_line e b c; f = on_line f c d, on_circle f z b; g = on_line g a e, on_circle g z b ? para a c f g
+
+user: Given acute triangle ABC, Draw the altitudes AD, BE, CF. Let H be the orthocenter of the triangle. Let M, N, P, Q be the perpendicular projections of D onto AB, BE, CF, AC, respectively. Prove: The four points M, N, P, Q are collinear.
+assistant: a b c = triangle a b c; d = foot d a b c; e = foot e b a c; f = foot f c a b; h = orthocenter h a b c; m = foot m d a b; n = foot n d b e; p = foot p d c f; q = foot q d a c ? coll m n p q
+
+user: Given triangle ABC inscribed in a circle with center O. D and E are the midpoints of arcs AB; AC respectively. Call the intersection of DE with AB; AC H and K respectively. Call I the intersection of BE with CD. Prove: AI is perpendicular to DE
+assistant: a b c = triangle a b c; o = circle o a b c; d = on_circle d o a, angle_bisector d a o b; e = on_circle e o a, angle_bisector e a o c; h = on_line h d e, on_line h a b; k = on_line k d e, on_line k a c; i = on_line i b e, on_line i c d ? perp a i d e
+
+user: Given ∆ABC is right-angled at A, with height AH. HD, HE are the bisectors of angles BHA and CHA respectively (D, E belong to AB, AC). I is the midpoint of DE. BI intersects DH, CD at M, P respectively; CI intersects EH, BE at N, Q respectively. BE intersects CD at K. Prove: MN // DE
+assistant: a b c = r_triangle a b c; h = foot h a b c; d = angle_bisector d b h a, on_line d a b; e = angle_bisector e c h a, on_line e a c; i = midpoint i d e; m = on_line m b i, on_line m d h; p = on_line p b i, on_line p c d; n = on_line n c i, on_line n e h; q = on_line q c i, on_line q b e; k = on_line k b e, on_line k c d ? para m n d e
+
+user: Given a right triangle ABC at A, height AH. Draw a circle with center A and radius AH. Let HD be the diameter of circle (A; AH). The tangent of the circle at D intersects CA at E. Let I be the projection of A on BE. Prove that AI = AH.
+assistant: a b c = r_triangle a b c; h = foot h a b c; d = on_circle d a h, on_line d a h; e = lc_tangent e d a, on_line e c a; i = foot a b e ? cong a i a h
+
+user: Given triangle ABC has three acute angles inscribed in a circle with center O and fixed radius R and D is the foot of the internal bisector of angle A of the triangle. Let E and F be the centers of the circumscribed circles of triangles ABD and ACD, respectively. Prove that quadrilateral AEOF is an inscribed quadrilateral.
+assistant: a b c = triangle a b c; o = circle o a b c; d = angle_bisector d b a c, on_line d b c; e = circle e a b d; f = circle f a c d ? cyclic a e o f
+
+user: Given triangle ABC has three acute angles and is inscribed in circle (O). Let H be the orthocenter of triangle ABC and P be a point on segment BC (P is different from B, C). Line AH intersects (BHC) at T different from H. Line PT intersects (BHC) at K different from T. Suppose BK intersects AC at M and CK intersects AB at N. Let X, Y be the midpoints of BN, CM respectively. Prove that quadrilateral ANKM is inscribed in a circle.
+assistant: a b c = triangle a b c; o = circle o a b c; h = orthocenter h a b c; p = on_line p b c; o1 = circle o1 b h c; t = on_line t a h, on_circle t o1 b; k = on_line k p t, on_circle k o1 b; m = on_line m b k, on_line m a c; n = on_line n c k, on_line n a b; x = midpoint x b n; y = midpoint y c m ? cyclic a n k m
+
+user: Given triangle ABC inscribed in a circle with center O. D and E are the midpoints of arcs AB; AC respectively. Call the intersections of DE with AB; AC H and K respectively. Prove: triangle AHK is isosceles.
+assistant: a b c = triangle a b c; o = circle o a b c; d = on_circle d o a, angle_bisector d a o b; e = on_circle e o a, angle_bisector e a o c; h = on_line h d e, on_line h a b; k = on_line k d e, on_line k a c ? cong a h a k
+
+user: Given an equilateral triangle ABC with height AH. On side BC, take any point M (M does not coincide with B, C, H); from M, draw MP, MQ perpendicular to sides AB. AC. Prove that OH ⊥ PQ
+assistant: b c = segment b c; a = eq_triangle a b c; h = foot h a b c; m = on_line m b c; p = foot m a b; q = foot m a c; o = midpoint o a m ? perp o h p q
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