2000_p6
a b c = triangle a b c; h = orthocenter h a b c; t1 t2 t3 i = incenter2 t1 t2 t3 i a b c; h1 = foot h1 a b c; h2 = foot h2 b c a; h3 = foot h3 c a b; x1 = reflect x1 h1 t1 t2; x2 = reflect x2 h2 t1 t2; y2 = reflect y2 h2 t2 t3; y3 = reflect y3 h3 t2 t3; z = on_line z x1 x2, on_line z y2 y3 | q = midpoint q b i; s = midpoint s i h2 ? cong i z i t1
2002_p2a
b c = segment b c; o = midpoint o b c; a = on_circle a o b; d = on_circle d o b, on_bline d a b; e = on_bline e o a, on_circle e o b; f = on_bline f o a, on_circle f o b; j = on_pline j o a d, on_line j a c ? eqangle e c e j e j e f
2002_p2b
b c = segment b c; o = midpoint o b c; a = on_circle a o b; d = on_circle d o b, on_bline d a b; e = on_bline e o a, on_circle e o b; f = on_bline f o a, on_circle f o b; j = on_pline j o a d, on_line j a c ? eqangle c e c j c j c f
2003_p4
a b c = triangle a b c; o = circle o a b c; b1 = on_circle b1 o a, on_bline b1 c a; d1 = on_circle d1 o a, on_bline d1 c a; x = on_line x b b1, on_line x a c; d = on_line d d1 x, on_circle d o a; p = foot p d b c; q = foot q d c a; r = foot r d a b ? cong p q q r
2004_p1
a b c = triangle a b c; o = midpoint o b c; m = on_circle m o b, on_line m a b; n = on_circle n o b, on_line n a c; r = angle_bisector r b a c, angle_bisector r m o n; o1 = circle o1 b m r; o2 = circle o2 c n r; p = on_circle p o1 r, on_circle p o2 r | k = on_bline k m n; l = eqdistance l k k a, eqdistance l o o a ? coll p b c
2004_p5
a b c = triangle a b c; o = circle o a b c; d = on_circle d o a; p = on_aline p b c a b d, on_aline p d c a d b ? cong a p c p
2005_p5
a b c = triangle a b c; d = eqdistance d a b c; e = on_line e b c; f = on_line f a d, eqdistance f d e b; p = on_line p a c, on_line p b d; q = on_line q e f, on_line q b d; r = on_line r e f, on_line r a c; o1 = circle o1 a p d; o2 = circle o2 b p c; m = on_circle m o1 p, on_circle m o2 p ? cyclic p q r m
2007_p4
a b c = triangle a b c; o = circle o a b c; r = on_circle r o a, on_bline r a b; l = midpoint l c a; k = midpoint k c b; p = on_line p o k, on_line p c r; q = on_line q o l, on_line q c r; l1 = foot l1 l c r; k1 = foot k1 k c r ? eqratio k k1 l l1 r q r p
2008_p1a
a b c = triangle a b c; h = orthocenter h a b c; d = midpoint d b c; e = midpoint e a c; f = midpoint f a b; a1 = on_circle a1 d h, on_line a1 b c; a2 = on_circle a2 d h, on_line a2 b c; b1 = on_circle b1 e h, on_line b1 c a; b2 = on_circle b2 e h, on_line b2 c a; c1 = on_circle c1 f h, on_line c1 a b; c2 = on_circle c2 f h, on_line c2 a b | o1 = eqdistance o1 e e h, on_tline o1 h e f ? cyclic c1 c2 b1 b2
2008_p1b
a b c = triangle a b c; h = orthocenter h a b c; d = midpoint d b c; e = midpoint e a c; f = midpoint f a b; a1 = on_circle a1 d h, on_line a1 b c; a2 = on_circle a2 d h, on_line a2 b c; b1 = on_circle b1 e h, on_line b1 c a; b2 = on_circle b2 e h, on_line b2 c a; c1 = on_circle c1 f h, on_line c1 a b; c2 = on_circle c2 f h, on_line c2 a b | o1 = eqdistance o1 e e h, on_tline o1 h e f; o2 = eqdistance o2 e e h, on_tline o2 h e d; o3 = eqdistance o3 f f h, on_tline o3 h f d; o = circle o c1 c2 b1 ? cyclic c1 c2 b1 b2 a1 a2
2008_p6
x@4.96_-0.13 y@-1.0068968328888160_-1.2534881080682770 z@-2.8402847238575120_-4.9117762734006830 = triangle x y z; o = circle o x y z; w@6.9090049230038776_-1.3884003936987552 = on_circle w o x; a = on_tline a z o z, on_tline a x o x; b = on_tline b z o z, on_tline b w o w; c = on_tline c y o y, on_tline c w o w; d = on_tline d x o x, on_tline d y o y; i1 = incenter i1 a b c; i2 = incenter i2 a c d; f1 = foot f1 i1 a c; f2 = foot f2 i2 a c; q t p s = cc_tangent q t p s i1 f1 i2 f2; k = on_line k q t, on_line k p s ? cong o k o x
2009_p2
m l k = triangle m l k; w = circle w m l k; q = on_tline q m w m; p = mirror p q m; b = mirror b p k; c = mirror c q l; a = on_line a b q, on_line a c p; o = circle o a b c | d = eqdistance d l m k, eqdistance d m l k; e = mirror e k w; f = mirror f q d ? cong o p o q
2010_p2
a b c = triangle a b c; o = circle o a b c; i = incenter i a b c; d = on_line d a i, on_circle d o a; f = on_line f b c; e = on_aline e a c b a f, on_circle e o a; g = midpoint g i f; k = on_line k d g, on_line k e i | h = mirror h e o; l = midpoint l a i; m = midpoint m b i ? cong o a o k
2010_p4
s c p = iso_triangle s c p; o = on_tline o c s c; a = on_circle a o c; b = on_circle b o c, on_line b s a; m = on_line m c p, on_circle m o c; l = on_line l b p, on_circle l o c; k = on_line k a p, on_circle k o c ? cong m k m l
2011_p6
a b c = triangle a b c; o = circle o a b c; p = on_circle p o a; q = on_tline q p o p; pa = reflect pa p b c; pb = reflect pb p c a; pc = reflect pc p a b; qa = reflect qa q b c; qb = reflect qb q c a; qc = reflect qc q a b; a1 = on_line a1 pb qb, on_line a1 pc qc; b1 = on_line b1 pa qa, on_line b1 pc qc; c1 = on_line c1 pa qa, on_line c1 pb qb; o1 = circle o1 a1 b1 c1; x = on_circle x o a, on_circle x o1 a1 ? coll x o o1
2012_p1
a b c = triangle a b c; m l k j = excenter2 m l k j a b c; f = on_line f m l, on_line f b j; g = on_line g m k, on_line g c j; s = on_line s f a, on_line s b c; t = on_line t g a, on_line t c b ? cong m s m t
2012_p5
c a b = r_triangle c a b; d = foot d c a b; x = on_line x c d; k = on_line k a x, on_circle k b c; l = on_line l b x, on_circle l a c; m = on_line m a l, on_line m b k | j = on_bline j a b, on_bline j c a; e = on_tline e d a j, on_tline e a b x ? cong m k m l
2013_p4
a b c = triangle a b c; h = orthocenter h a b c; m = on_line m h b, on_line m a c; n = on_line n h c, on_line n a b; w = on_line w b c; o1 = circle o1 b n w; o2 = circle o2 c m w; x = on_line x o1 w, on_circle x o1 w; y = on_line y o2 w, on_circle y o2 w ? coll x h y
2014_p4
a b c = triangle a b c; p = on_line p b c, on_aline p a b b c a; q = on_line q b c, on_aline q a c c b a; m = mirror m a p; n = mirror n a q; x = on_line x b m, on_line x c n; o = circle o a b c ? cong o x o a
2015_p3
a b c = triangle a b c; h = orthocenter h a b c; f = on_line f h a, on_line f b c; m = midpoint m b c; o = circle o a b c; q = on_dia q a h, on_circle q o a; k = on_dia k h q, on_circle k o a; o1 = circle o1 k q h; o2 = circle o2 f k m ? coll o1 o2 k
2015_p4
a b c = triangle a b c; o = circle o a b c; d = on_line d b c; e = on_line e b c, on_circle e a d; f = on_circle f o a, on_circle f a d; g = on_circle g o a, on_circle g a d; o1 = circle o1 f b d; o2 = circle o2 g c e; k = on_circle k o1 b, on_line k a b; l = on_circle l o2 c, on_line l a c; x = on_line x f k, on_line x l g ? coll x o a
2016_p1
a b z = triangle a b z; f = angle_bisector f b a z, on_bline f a b; c = on_tline c b f b, on_line c a f; d = on_line d a z, on_bline d a c; e = angle_mirror e c a d, on_bline e a d; m = midpoint m c f; x = parallelogram x e a m; y = on_line y f x, on_line y e m ? coll y b d
2017_p4
r s = segment r s; t = mirror t r s; o = on_bline o r s; j = on_circle j o s; o1 = circle o1 j s t; a = on_tline a r o r, on_circle a o1 s; b = on_tline b r o r, on_circle b o1 s; k = on_line k j a, on_circle k o s ? perp k t o1 t
2018_p1
a b c = triangle a b c; o = circle o a b c; d = on_line d a b; e = on_line e a c, on_circle e a d; f = on_bline f b d, on_circle f o a; g = on_bline g e c, on_circle g o a | i = on_circle i d a, on_circle i f a; j = on_circle j e a, on_circle j g a ? para d e f g
2019_p2
a b c = triangle a b c; a1 = on_line a1 b c; b1 = on_line b1 a c; p = on_line p a a1; q = on_line q b b1, on_pline q p a b; p1 = on_line p1 p b1, eqangle3 p1 p c a b c; q1 = on_line q1 q a1, eqangle3 q1 c q b c a | o = circle o a b c; a2 = on_circle a2 o a, on_line a2 a a1; b2 = on_circle b2 o b, on_line b2 b b1 ? cyclic p q p1 q1
2019_p6
a b c = triangle a b c; d e f i = incenter2 d e f i a b c; r = on_tline r d e f, on_circle r i d; p = on_line p r a, on_circle p i d; o1 = circle o1 p c e; o2 = circle o2 p b f; q = on_circle q o1 p, on_circle q o2 p; t = on_line t p q, on_line t i d | n = orthocenter n c a i; g = on_pline g c a b, on_pline g a c n ? perp a t a i
2020_p1
p a b = triangle p a b; x = angle_bisector x p b a; y = angle_bisector y p a b; z = on_aline z a p a b x; t = on_aline t p a p a z; d = on_aline d p t p b a, on_line d a z; u = on_aline u b p b a y; v = on_aline v p b p b u; c = on_aline c p v p a b, on_line c b u; o = angle_bisector o a d p, angle_bisector o p c b | j = intersection_ll j o c a y; k = reflect k p j c ? cong o a o b
2021_p3
a b c = triangle a b c; d = angle_bisector d b a c; e = on_aline e d a d c b, on_line e a c; f = on_aline f d a d b c, on_line f a b; x = on_bline x b c, on_line x a c; o1 = circle o1 a d c; o2 = circle o2 e x d; y = on_line y e f, on_line y b c ? coll o1 o2 y
2022_p4
b c = segment b c; d = free d; e = eqdistance e d b c; t = on_bline t b d, on_bline t c e; a = eqangle2 a b t e; p = on_line p a b, on_line p c d; q = on_line q a b, on_line q c t; r = on_line r a e, on_line r c d; s = on_line s a e, on_line s d t ? cyclic p q r s
1975_p3
a b c = triangle a b c; p = s_angle c b p 45o, s_angle b c p 150o; q = s_angle a c q 30o, s_angle c a q 135o; r = s_angle b a r 15o, s_angle a b r 165o | t = eq_triangle t b r ? perp q r r p; cong q r r p
1983_p2
a o1 o2 = triangle a o1 o2; p1 p2 q1 q2 = cc_tangent p1 p2 q1 q2 o1 a o2 a; m1 = midpoint m1 p1 q1; m2 = midpoint m2 p2 q2 ? eqangle a o1 a o2 a m1 a m2
1995_p1
a b = segment a b; c = on_line c a b; d = on_line d a b; o1 = midpoint o1 a c; o2 = midpoint o2 b d; x = on_circle x o1 a, on_circle x o2 b; y = on_circle y o1 a, on_circle y o2 b; z = on_line z x y, on_line z b c; p = on_line p x y; m = on_circle m o1 a, on_line m c p; n = on_line n b p, on_circle n o2 b; q = on_line q a m, on_line q d n ? coll q x y
2024_p4
a b c = triangle a b c; t1 t2 t3 i = incenter2 t1 t2 t3 i a b c; o = circle o a b c; s2 = mirror s2 t2 i; x = on_line x b c, on_pline x s2 a c; s3 = mirror s3 t3 i; y = on_line y b c, on_pline y s3 a b; p = on_line p a i, on_circle p o a; k = midpoint k a c; l = midpoint l a b ? eqangle k i i l x p p y
2025_p2
a m n = triangle a m n; b = on_circle b m a, on_circle b n a; c = on_line c m n, on_circle c m a; d = on_line d m n, on_circle d n a; p = circle p a c d; e = on_line e a p, on_circle e m a; f = on_line f a p, on_circle f n a; h = on_tline h p m n, on_tline h m p n; o = circle o b e f; x = on_dia x o h, on_pline x h a p | w = circle w d e f ? cong o x o b; obtuse_angle c m n; obtuse_angle m n d