imo_sl_1998_g1_part1
a b c = triangle a b c; d = on_circum d a b c, on_tline d b a c; p = on_bline p a b, on_bline p c d; f1 = foot f1 a b p; f2 = foot f2 c d p ? eqratio a f1 c f2 d p b p
imo_sl_1999_g6
p m q = iso_triangle p m q; o = on_line o p m; n = on_line n o q, on_circle n o m; x = on_circle x p q, on_circle x q n; y = on_circle y p q, on_circle y q n; a = on_line a x y, on_circle a o m; b = on_line b x y, on_circle b o m; c = on_line c a m, on_circle c p m; d = on_line d b m, on_circle d p m; t = on_line t c d, on_circle t q n ? perp c t t q
imo_sl_1999_g6_oracle
p m q = iso_triangle p m q; o = on_line o p m; n = on_line n o q, on_circle n o m; x = on_circle x p q, on_circle x q n; y = on_circle y p q, on_circle y q n; a = on_line a x y, on_circle a o m; b = on_line b x y, on_circle b o m; c = on_line c a m, on_circle c p m; d = on_line d b m, on_circle d p m; t = on_line t p q, on_circle t q n ? perp c t t q
imo_sl_2000_g1
o1 x y = iso_triangle o1 x y; o2 = on_bline o2 x y; z = on_line z x y; p = on_dia p z o1, on_circle p o1 x; q = on_dia q z o2, on_circle q o2 x; o3 = on_line o3 o1 p, on_line o3 o2 q; r = on_line r x y, on_circle r o3 p; t1 t2 t3 t4 = cc_tangent t1 t2 t3 t4 o1 x o2 x ? coll p r t1
imo_sl_2000_g2
a b = segment a b; g1 = on_tline g1 a a b; g2 = on_tline g2 b b a; m = on_circle m g1 a, on_circle m g2 b; n = on_circle n g1 a, on_circle n g2 b; c = on_pline c m a b, on_circle c g1 a; d = on_pline d m a b, on_circle d g2 b; e = on_line e a c, on_line e b d; p = on_line p a n, on_line p c d; q = on_line q b n, on_line q c d ? cong e p e q
imo_sl_2000_g5
a b c = triangle a b c; o = circle o a b c; b1 = on_tline b1 a o a, on_tline b1 c o c; e = intersection_ll e b b1 a c; n = midpoint n b e; a1 = on_tline a1 b o b, on_tline a1 c o c; d = intersection_ll d a a1 b c; m = midpoint m a d ? eqangle a b b m n a a b
imo_sl_2000_g6
x b c d = quadrangle x b c d; a = on_aline a d x x c b, eqangle3 a d x b x c; h = foot h b x c; y = on_aline y b a c b h, on_bline y a b ? cong y c y d
imo_sl_2000_g8
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 ? cong i z i t1
imo_sl_2002_g1
o b d = iso_triangle o b d; a = lc_tangent a b o; w = on_line w o d; c x = tangent c x a w d; u = circle u b c d ? cyclic u a b c
imo_sl_2002_g3a
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
imo_sl_2002_g3b
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
imo_sl_2002_g4
o1 p q = iso_triangle o1 p q; o2 = on_bline o2 p q; a1 = on_circle a1 o1 p; b1 = on_circle b1 o1 p; a2 = intersection_lc a2 a1 o2 p; b2 = intersection_lc b2 b1 o2 p; c = intersection_ll c a1 b1 a2 b2; o3 = circle o3 a1 a2 c ? cyclic o1 o2 o3 q
imo_sl_2002_g7
a b c = triangle a b c; i = incenter i a b c; k = on_tline k i b c, on_line k b c; d = foot d a b c; m = midpoint m a d; n = intersection_lc n m i k; o = circle o b c n; x = lc_tangent x n i ? perp o n n x
imo_sl_2002_g7_encoding2
a b c = triangle a b c; i = incenter i a b c; k = on_tline k i b c, on_line k b c; d = foot d a b c; m = midpoint m a d; n = on_line n k m, on_circle n i k; s = circle s b c n ? coll n i s
imo_sl_2002_g8
a b = segment a b; c = free c; y = free y; o1 = circle o1 a b c; o2 = circle o2 a b y; d = intersection_lc d c o2 a; m = on_line m c d; n = on_pline n m b d, on_line n b c; k = on_pline k m b c, on_line k b d; e = on_tline e n b c, on_circle e o1 a; f = on_tline f k b d, on_circle f o2 a ? perp e m m f
imo_sl_2002_g8_encoding2
a b = segment a b; c = free c; d = on_line d a c; m = on_line m c d; n = on_pline n m b d, on_line n b c; k = on_pline k m b c, on_line k b d; e = on_tline e n b c, on_circum e a b c; f = on_tline f k b d, on_circum f a b d ? perp e m m f
imo_sl_2003_g1a
p r = segment p r; q = midpoint q p r; d = free d; a = on_tline a q d q, on_tline a r d r; b = on_tline b p d p, on_tline b r d r; c = on_tline c p d p, on_tline c q d q; e = angle_bisector e a b c, on_line e a c ? eqangle a d d e e d d c
imo_sl_2003_g1b
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
imo_sl_2003_g2
b a c = between b a c; d = free d; o = circle o a c d; p = on_tline p a a o, on_tline p c c o; q = on_line q p b, on_circle q o a; r = angle_bisector r a q c, on_line r a c; d1 = free d1; o1 = circle o1 a c d1; p1 = on_tline p1 a a o1, on_tline p1 c c o1; q1 = on_line q1 p1 b, on_circle q1 o1 a; s = on_line s r q, on_circle s o a; s1 = on_line s1 r q1, on_circle s1 o1 a ? eqangle a q1 q1 r r q1 q1 c
imo_sl_2003_g5
c a b = iso_triangle c a b; i = incenter i c a b; o = circle o a i b; p = on_circle p o i; d = on_pline d p c a, on_line d a b; e = on_pline e p c b, on_line e a b; f = on_pline f p a b, on_line f c a; g = on_pline g p a b, on_line g c b; x = intersection_ll x d f e g ? cyclic x a b c
imo_sl_2004_g1
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 ? coll p b c
imo_sl_2004_g2
m d a = r_triangle m d a; b = on_line b a m; o = midpoint o a b; c = on_circle c o a, on_line c a d; e = on_circle e o a, on_dia e o d; f = intersection_ll f b e m d; g = intersection_lc g f o a; h = reflect h g a b ? coll h c f
imo_sl_2004_g3b
c x y = ieq_triangle c x y; b = on_circle b x c; a = on_line a x b, on_line a y c; o = circle o a b c; d = intersection_ll d a o b c; e = circle e a b d; f = circle f a c d; g = on_line g a b, on_circle g a c; h = on_line h a c, on_circle h a b ? perp e f f g
imo_sl_2004_g3c
c x y = ieq_triangle c x y; b = on_circle b x c; a = on_line a x b, on_line a y c; o = circle o a b c; d = intersection_ll d a o b c; e = circle e a b d; f = circle f a c d; g = on_line g a b, on_circle g a c; h = on_line h a c, on_circle h a b ? perp f g g h
imo_sl_2004_g4a
a b c = triangle a b c; d = on_circum d a b c; p = on_aline p b c a b d, on_aline p d c a d b ? cong a p c p
imo_sl_2004_g7
a b c = triangle a b c; x = on_line x c b; d e f i = incenter2 d e f i a b x; d' e' f' j = incenter2 d' e' f' j a c x; p = on_circle p i d, on_circle p j d'; q = intersection_cc q i j p; m = midpoint m a b; n = midpoint n a c; u = on_line u m n, angle_bisector u a b c; v = on_line v m n, angle_bisector v a c x; w = midpoint w u v ? coll p q w
imo_sl_2004_g8
a b c = triangle a b c; o = circle o a b c; d = on_circle d o a; e = intersection_ll e a d b c; f = intersection_ll f a c b d; m = midpoint m c d; o1 = circle o1 a b m; n = eqratio6 n a b m a m b, on_circle n o1 a ? coll e f n
imo_sl_2005_g1
a c = segment a c; p1 = on_line p1 c a, eqdistance p1 c c a; p2 = on_line p2 p1 c, eqdistance p2 p1 c a; x = between_bound x a p2; b = eqdistance b a a x, eqdistance b c p2 x; i = incenter i c a b; d = foot d i a b; e = foot e i b c; k = mirror k d i; l = mirror l e i ? cyclic a c k l
imo_sl_2005_g4
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
imo_sl_2005_g5
a d e = iso_triangle a d e; h = on_line h d e; x = foot x h a d; y = foot y h a e; c = intersection_ll c x h a e; b = intersection_ll b a d h y; m = midpoint m b c; o1 = circle o1 a b c; o2 = circle o2 a d e; z = on_circle z o1 a, on_circle z o2 a ? perp h m a z
imo_sl_2005_g6
a b c = triangle a b c; m = midpoint m b c; d e f i = incenter2 d e f i a b c; k = on_circle k i e, on_line k a m; l = intersection_lc l a i k; x = on_pline x k b c, on_circle x i e; y = on_pline y l b c, on_circle y i e; p = intersection_ll p a x b c; q = intersection_ll q a y b c ? cong b p c q
imo_sl_2006_g2
a b c d = trapezoid a b c d; k = between_bound k a b; l = eqratio6 l d c k a k b, on_line l d c; p = on_line p k l, eqangle3 p a b c d b; q = on_line q k l, eqangle3 q c d b c a ? cyclic p q b c
imo_sl_2006_g3
a b c = triangle a b c; d = on_aline d a c c a b, on_aline d c a c b a; e = on_aline e a d d a c, on_aline e d a d c a; p = intersection_ll p b d c e; m = intersection_ll m a p c d ? cong c m m d
imo_sl_2006_g4
a b c = triangle a b c; d = eqdistance d b b a, on_line d a c; x l k i = incenter2 x l k i a b c; j = incenter j b c d; p = intersection_ll p a j k l ? cong a p p j
imo_sl_2006_g5a
j a1 b1 = iso_triangle j a1 b1; c1 = on_circle c1 j a1, on_pline c1 j a1 b1; c = on_tline c a1 a1 j, on_tline c b1 b1 j; a = on_tline a b1 b1 j, on_tline a c1 c1 j; b = on_tline b a1 a1 j, on_tline b c1 c1 j; d = intersection_ll d a b a1 b1; e = foot e c1 d j ? perp b e e a1
imo_sl_2006_g5b
j a1 b1 = iso_triangle j a1 b1; c1 = on_circle c1 j a1, on_pline c1 j a1 b1; c = on_tline c a1 a1 j, on_tline c b1 b1 j; a = on_tline a b1 b1 j, on_tline a c1 c1 j; b = on_tline b a1 a1 j, on_tline b c1 c1 j; d = intersection_ll d a b a1 b1; e = foot e c1 d j ? perp a e e b1
imo_sl_2006_g6
o1 o2 o = triangle o1 o2 o; d e f i = excenter2 d e f i o o2 o1; a = on_circle a o e, on_pline a o o1 o2; b = on_circle b o e, on_pline b o o1 o2; x = intersection_ll x a o1 b o2 ? coll x e f
imo_sl_2006_g7a
a b c = triangle a b c; o = circle o a b c; ma = midpoint ma b c; mb = midpoint mb a c; mc = midpoint mc a b; ta = angle_bisector ta c a b, on_circle ta o a; tb = angle_bisector tb a b c, on_circle tb o a; tc = angle_bisector tc b c a, on_circle tc o a; oa = midpoint oa ma ta; ob = midpoint ob mb tb; oc = midpoint oc mc tc; xc yc xc' yc' = cc_tangent xc yc xc' yc' oa ma ob mb; xb' yb' xb yb = cc_tangent xb' yb' xb yb oc mc oa ma; xa ya xa' ya' = cc_tangent xa ya xa' ya' ob mb oc mc; a' = intersection_ll a' xb yb xc yc; b' = intersection_ll b' xa ya xc yc; c' = intersection_ll c' xa ya xb yb ? simtri a b c a' b' c'
imo_sl_2006_g7b
a b c = triangle a b c; o = circle o a b c; ma = midpoint ma b c; mb = midpoint mb a c; mc = midpoint mc a b; ta = angle_bisector ta c a b, on_circle ta o a; tb = angle_bisector tb a b c, on_circle tb o a; tc = angle_bisector tc b c a, on_circle tc o a; oa = midpoint oa ma ta; ob = midpoint ob mb tb; oc = midpoint oc mc tc; xc yc xc' yc' = cc_tangent xc yc xc' yc' oa ma ob mb; xb' yb' xb yb = cc_tangent xb' yb' xb yb oc mc oa ma; xa ya xa' ya' = cc_tangent xa ya xa' ya' ob mb oc mc; a' = intersection_ll a' xb yb xc yc; b' = intersection_ll b' xa ya xc yc; c' = intersection_ll c' xa ya xb yb; na = midpoint na mb mc; nb = midpoint nb ma mc; nc = midpoint nc ma mb ? contri na nb nc a' b' c'
imo_sl_2006_g9
a b c = triangle a b c; o = circle o a b c; a1 = between_bound a1 b c; b1 = between_bound b1 a c; c1 = between_bound c1 a b; o1 = circle o1 a1 b1 c; o2 = circle o2 a1 c1 b; o3 = circle o3 b1 c1 a; a2 = intersection_cc a2 o3 o a; b2 = intersection_cc b2 o2 o b; c2 = intersection_cc c2 o1 o c; m1 = midpoint m1 b c; m2 = midpoint m2 c a; m3 = midpoint m3 a b; a3 = mirror a3 a1 m1; b3 = mirror b3 b1 m2; c3 = mirror c3 c1 m3 ? simtri a2 b2 c2 a3 b3 c3
imo_sl_2007_g1
a b c = triangle a b c; o = circle o a b c; r = angle_bisector r a c b, on_circle r o a; t = midpoint t a c; s = midpoint s b c; p = on_line p c r, on_bline p b c; q = on_line q c r, on_bline q a c; x = foot x t c r; y = foot y s c r ? eqratio x t y s p r q r
imo_sl_2007_g2
a b c = iso_triangle a b c; m = midpoint m b c; x = on_circum x a b m; t = on_tline t m m x, on_circle t x b; n = midpoint n t b; u = on_aline u t m m t c; x1 = on_circum x1 a b m; t1 = on_tline t1 m m x1, on_circle t1 x1 b; n1 = midpoint n1 t1 b; u1 = on_aline u1 t1 m m t1 c ? eqangle u t t b u1 t1 t1 b
imo_sl_2007_g2_cheated
a b c = iso_triangle a b c; m = midpoint m b c; x = on_circum x a b m; t = on_tline t m m x, on_circle t x b; u = on_aline u t m b a m; n = midpoint n t b ? eqangle u t t b c t t m
imo_sl_2007_g2_encoding2
a b c = iso_triangle a b c; m = midpoint m b c; o1 = circle o1 a b m; x = on_circle x o1 a; t = on_tline t m m x, on_circle t x b; u = on_aline u t m m t c; x1 = on_circle x1 o1 a; t1 = on_tline t1 m m x1, on_circle t1 x1 b; u1 = on_aline u1 t1 m m t1 c ? eqangle u t t b u1 t1 t1 b
imo_sl_2007_g3
a q d = triangle a q d; c = free c; b = on_pline b c a d, on_aline b q c d q a; p = intersection_ll p a c b d ? eqangle b q q p d a a q
imo_sl_2007_g8
c p d = triangle c p d; k l x i = incenter2 k l x i c d p; ia = on_line ia k i; y = on_dia y d ia, on_circle y ia k; z = on_dia z p ia, on_circle z ia k; a = on_line a p z, on_line a d y; t = on_line t p a, eqdistance t p p l; ib = angle_bisector ib t p c, on_line ib i l; u = on_dia u c ib, on_circle u ib l; b = on_line b c u, on_line b a p; e = intersection_ll e a c b d; f = intersection_ll f a k b l ? coll e i f
imo_sl_2008_g1a
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; o = on_circle o e h, on_tline o h e f ? cyclic c1 c2 b1 b2
imo_sl_2008_g1a_encoding1
a b c = triangle a b c; h = orthocenter h a b c; e = midpoint e a c; f = midpoint f a b; 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; d = midpoint d b c; a1 = on_circle a1 d h, on_line a1 b c; a2 = on_circle a2 d h, on_line a2 b c ? cyclic c1 c2 b1 b2
imo_sl_2008_g1a_encoding2
a b c = triangle a b c; h = orthocenter h a b c; e = midpoint e a c; f = midpoint f a b; 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 ? cyclic c1 c2 b1 b2
imo_sl_2008_g1b
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 ? cyclic c1 c2 b1 a1
imo_sl_2008_g3
e p q = between e p q; a = free a; d = on_circum d a p q, eqangle3 d e q a p e; b = free b; c = on_circum c b p q, eqangle3 c e q b p e ? cyclic a b c d
imo_sl_2008_g7
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
imo_sl_2009_g2
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 ? cong o p o q
imo_sl_2009_g3
a b c = triangle a b c; x y z i = incenter2 x y z i a b c; g = intersection_ll g b y c z; r = parallelogram r b c y; s = parallelogram s z b c ? cong g r g s
imo_sl_2009_g4
a b c = triangle a b c; d = on_circum d a b c; e = intersection_ll e a c b d; f = intersection_ll f a d b c; g = midpoint g a b; h = midpoint h c d; w = circle w e g h ? perp w e e f
imo_sl_2009_g6
a b c d = quadrangle a b c d; p = intersection_ll p a d b c; o1 = circle o1 a b p; o2 = circle o2 d c p; h1 = orthocenter h1 a b p; h2 = orthocenter h2 d c p; e1 = midpoint e1 o1 h1; e2 = midpoint e2 o2 h2; x = on_tline x e1 c d, on_tline x e2 a b ? coll x h1 h2
imo_sl_2009_g8
x y z = triangle x y z; o = circle o x y z; t = on_circle t o x; a = on_tline a x o x, on_tline a t o t; b = on_tline b y o y, on_tline b x o x; c = on_tline c z o z, on_tline c y o y; d = on_tline d t o t, on_tline d z o z; m = on_line m b c; n = intersection_ll n a m c d; i1 = incenter i1 a b m; i2 = incenter i2 m n c; i3 = incenter i3 n d a; h = orthocenter h i1 i2 i3 ? coll a m h
imo_sl_2010_g1
a b c = triangle a b c; d = foot d a b c; e = foot e b c a; f = foot f c a b; p = on_circum p a b c, on_line p e f; q = intersection_ll q b p d f ? cong a p a q
imo_sl_2010_g2a
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
imo_sl_2010_g2b
m k l = iso_triangle m k l; o = circle o k l m; p = free p; a = intersection_lc a p o k; b = intersection_lc b p o l; c = intersection_lc c p o m; s = on_tline s c c o, on_line s a b ? cong s c s p
imo_sl_2010_g4
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 ? cong o a o k
imo_sl_2011_g3
a b c d = quadrangle a b c d; o1 = midpoint o1 a b; o2 = midpoint o2 c d; e = on_circle e o1 a, on_circle e o2 c; f = intersection_cc f o1 o2 e; xe = foot xe e a b; ye = foot ye e b c; ze = foot ze e c d; oe = circle oe xe ye ze; xf = foot xf f c d; yf = foot yf f d a; zf = foot zf f a b; of = circle of xf yf zf; m = midpoint m e f; u = on_circle u oe xe, on_circle u of xf; v = intersection_cc v oe of u ? coll m u v
imo_sl_2011_g4
w o x = between w o x; a = on_circle a o x; m = midpoint m a o; n = midpoint n a x; b0 = on_circle b0 w x, on_circle b0 m n; c0 = on_circle c0 w x, on_circle c0 m n; b = mirror b a b0; c = mirror c a c0; d = foot d a b c; g = intersection_ll g b c0 c b0 ? coll d g x
imo_sl_2011_g5
a b c = triangle a b c; o = circle o a b c; i = incenter i a b c; d = intersection_lc d i o a; e = intersection_lc e i o b; f = intersection_ll f d e a c; g = intersection_ll g d e b c; p = intersection_pp p f a d g b e; k = on_tline k a o a, on_tline k b o b; x = intersection_ll x a e b d ? coll x k p
imo_sl_2011_g6a
a b c = iso_triangle a b c; d = midpoint d a c; e = angle_bisector e b a c, on_circum e b c d; f = on_line f b d, on_circum f a e b; i = intersection_ll i a f b e; k = intersection_ll k c i b d ? eqangle k a a i i a a b
imo_sl_2011_g6c
a b c = iso_triangle a b c; d = midpoint d a c; e = angle_bisector e b a c, on_circum e b c d; f = on_line f b d, on_circum f a e b; i = intersection_ll i a f b e; k = intersection_ll k c i b d ? eqangle a b b i i b b k
imo_sl_2011_g7
x o a = r_triangle x o a; b = on_line b a x; y = on_dia y o b, on_circle y o x; c = on_line c b y, on_circle c o a; z = on_dia z o c, on_circle z o x; d = on_line d c z; t = on_dia t o d, on_circle t o x; e = on_line e d t, on_circle e o a; u = on_dia u o e, on_circle u o x; v = on_dia v o a, on_circle v o x; f = intersection_ll f a v e u; j = foot j b c d; k = on_tline k b d f, on_line k e o; l = foot l k d e ? cong d j d l
imo_sl_2011_g8
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
imo_sl_2012_g1
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
imo_sl_2012_g2
a b c = triangle a b c; d = on_circum d a b c; e = intersection_ll e a c b d; f = intersection_ll f a d b c; g = parallelogram g d e c; h = reflect h e a d ? eqangle d h h g d f f g
imo_sl_2012_g3
a b c = triangle a b c; d = foot d a b c; e = foot e b c a; f = foot f c a b; i1 = incenter i1 a e f; i2 = incenter i2 b d f; o1 = circle o1 a c i1; o2 = circle o2 b c i2 ? para i1 i2 o1 o2
imo_sl_2012_g4
a b c = triangle a b c; o = circle o a b c; d = angle_bisector d c a b, on_line d b c; m = midpoint m b c; e = mirror e d m; x = intersection_lt x a o d b c; y = intersection_lt y a d e b c ? cyclic b x c y
imo_sl_2012_g5
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 ? cong m k m l
imo_sl_2012_g6
a b c = triangle a b c; o = circle o a b c; i = incenter i a b c; d = on_line d b c; x = on_line x a b, on_circle x b d; f = on_line f a b, eqdistance f x a c; y = on_line y a c, on_circle y c d; e = on_line e a c, eqdistance e y a b; p = on_circum p b f d, on_circum p c d e ? cong o p o i
imo_sl_2012_g8
a b c = triangle a b c; o = circle o a b c; x = on_line x b c; y = on_line y c a; z = intersection_ll z x y a b; p = foot p o x y; q = on_circum q a x p, on_circum q b y p ? eqangle q c c p q z z p
imo_sl_2013_g1
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
imo_sl_2013_g2
a b c = triangle a b c; o = circle o a b c; m = midpoint m a b; n = midpoint n a c; t = angle_bisector t b a c, on_circle t o a; x = on_circum x a m t, on_bline x a c; y = on_circum y a n t, on_bline y a b; k = intersection_ll k m n x y ? cong k a k t
imo_sl_2013_g4
a b c = triangle a b c; p = on_line p a c, on_aline p b a b c a; q = on_line q a c, on_aline q b a a c b; d = on_line d b q, on_circle d p b; o = circle o a b c; r = intersection_lc r d o a ? cong q b q r
imo_sl_2014_g1
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
imo_sl_2014_g3
a b c = triangle a b c; o = circle o a b c; m = on_circle m o a, angle_bisector m a b c; w = midpoint w b m; p = on_circle p w b, angle_bisector p a o b; q = on_circle q w b, angle_bisector q b o c; r = on_line r p q, on_bline r b m ? para b r a c
imo_sl_2014_g4
a b c = triangle a b c; p = on_circum p a b c; d = on_line d a b; x = eqdistance x b c p; y = intersection_lp y b x d a x; m = on_line m c p, eqdistance m c y b; q = on_circum q a m p, on_circum q b m c ? eqangle d q q b a b b c
imo_sl_2014_g5
a c = segment a c; b = on_dia b a c; d = on_dia d a c; h = foot h a b d; k = on_line k a b, on_bline k h c; s = on_circle s k h, on_line s a b; l = on_line l a d, on_bline l h c; t = on_circle t l h, on_line t a d; o = circle o t s h; m = intersection_ll m k l h c; w = midpoint w a c; p1 = foot p1 a k l ? perp o h b d
imo_sl_2015_g1
a b c = triangle a b c; h = orthocenter h a b c; g = parallelogram g h a b; x = intersection_ll x a c g h; i = mirror i h x; o = circle o g c i; j = intersection_lc j a o c ? cong i j a h
imo_sl_2015_g2
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
imo_sl_2015_g3
c a b = r_triangle c a b; h = foot h c a b; x = on_line x c h; d = mirror d a x; p = intersection_ll p b d c h; o = midpoint o b d; q y = tangent q y p o b; z = intersection_ll z c q a d ? cong o z o b
imo_sl_2015_g5
a b c = triangle a b c; d = midpoint d a b; f = midpoint f a c; g = midpoint g b c; o = on_tline o d a b, on_bline o c d; h = on_circle h o d, on_line h a f; i = on_circle i o d, on_line i b g; h1 = mirror h1 h f; i1 = mirror i1 i g; q = intersection_ll q h1 i1 c d; m = intersection_ll m h1 i1 f g; p = intersection_lc p m o c ? cong c q q p
imo_sl_2015_g6
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
imo_sl_2016_g1
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
imo_sl_2016_g2
a b c = triangle a b c; o = circle o a b c; m = midpoint m b c; i = incenter i a b c; d = foot d i b c; f = on_tline f i a i, on_line f a b; e = on_tline e i a i, on_line e a c; w = circle w a e f; x = intersection_cc x o w a; y = intersection_ll y x d a m ? cong o y o a
imo_sl_2016_g4
a b c = iso_triangle a b c; i = incenter i a b c; d = intersection_ll d b i a c; e = intersection_lt e a i d a c; j = reflect j i a c; o = circle o b d e ? cong o j o b
imo_sl_2016_g5
a b c = triangle a b c; o = circle o a b c; m n k v = centroid m n k v a b c; d = foot d a o v; s = on_bline s a d; x = intersection_lc x b s a; y = intersection_lc y c s a; p = foot p a b c; w = circle w x s y ? cong w p w m
imo_sl_2016_g6
a b c = triangle a b c; d = eqangle3 d a c b c a; p = angle_bisector p a b c, angle_bisector p a d c; e = intersection_ll e b p a c; f = intersection_ll f d p a c; m = midpoint m a c; o = circle o b p d; x = intersection_lc x m o b; y = intersection_lc y m o d; q = intersection_ll q x e y f ? perp p q a c
imo_sl_2016_g7a
a b c = triangle a b c; i = incenter i a b c; ia = excenter ia a b c; ia1 = reflect ia1 ia b c; ia2 = reflect ia2 ia1 a i; ib = excenter ib b c a; ib1 = reflect ib1 ib c a; ib2 = reflect ib2 ib1 b i; p = intersection_ll p a ia2 b ib2; o = circle o a b c ? coll p o i
imo_sl_2017_g1
b a c = iso_triangle b a c; d = eqdistance d c b c; e = on_aline e a b b c d, on_aline e d c c b a; x = intersection_ll x a c b d ? perp e x b c
imo_sl_2017_g2
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
imo_sl_2017_g3
a b c = triangle a b c; o = circle o a b c; h = orthocenter h a b c; p = intersection_ll p o a b h; q = intersection_ll q o a c h; w = circle w p q h; m = midpoint m b c ? coll a m w
imo_sl_2017_g4
a b c = triangle a b c; d e f w = excenter2 d e f w a b c; o = circle o a e f; p = on_circle p o a, on_line p b c; q = intersection_lc q b o p; m = midpoint m a d; u = circle u m p q; x = on_circle x u m, on_circle x w e ? coll u w x
imo_sl_2017_g5
b a c = iso_triangle b a c; p = free p; q = on_tline q p a p, on_pline q b p a; a1 = reflect a1 a p q; b1 = reflect b1 b p q; c1 = reflect c1 c p q; d = intersection_ll d a c1 a1 c; o = circle o a b c; w = circle w a1 b c1; e = intersection_cc e o w b; x = intersection_ll x b b1 d e ? cong o x o a
imo_sl_2017_g7
p1 a b = between p1 a b; i = on_tline i p1 p1 a; p2 = on_dia p2 i b, on_circle p2 i p1; c = on_line c p2 b; p3 = on_dia p3 c i, on_circle p3 i p1; p4 = on_dia p4 a i, on_circle p4 i p1; d = on_line d p4 a, on_line d p3 c; ia = incenter ia d a b; ib = incenter ib a b c; ic = incenter ic b c d; id = incenter id c d a; oa = circle oa a ib id; oc = circle oc c ib id; p5 p6 p7 p8 = cc_tangent p5 p6 p7 p8 oa a oc c; x = on_line x p5 p6, on_line x p7 p8; ob = circle ob b ia ic; od = circle od d ia ic; p9 p10 p11 p12 = cc_tangent p9 p10 p11 p12 ob b od d; y = on_line y p9 p10, on_line y p11 p12 ? perp x i i y
imo_sl_2018_g1
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 ? para d e f g
imo_sl_2018_g2
a b c = iso_triangle a b c; m = midpoint m b c; p = on_pline p a b c; x = on_line x p b; y = on_line y p c, eqangle3 y m p x p m ? cyclic a p x y
imo_sl_2018_g4
a b c = triangle a b c; t = free t; a1 = reflect a1 t b c; b1 = reflect b1 t c a; c1 = reflect c1 t a b; o = circle o a1 b1 c1; a2 = intersection_lc a2 t o a1; b2 = intersection_lc b2 t o b1; x = intersection_lc x a o a2 ? coll x b b2
imo_sl_2018_g6
a b c = triangle a b c; d = eqratio6 d a c b a b c; e = intersection_ll e a b c d; f = intersection_ll f b c d a; x = on_circum x a e c, on_circum x b f d ? eqangle a x x b d x x c
imo_sl_2018_g7a
a b c = triangle a b c; o = circle o a b c; p = on_circle p o a; oa = circle oa o a p; ob = circle ob o b p; oc = circle oc o c p; x = on_tline x ob c a, on_tline x oc a b; y = on_tline y oc a b, on_tline y oa b c; z = on_tline z oa b c, on_tline z ob c a; w = circle w x y z ? cong w p w x
imo_sl_2018_g7b
a b c = triangle a b c; o = circle o a b c; p = on_circle p o a; oa = circle oa o a p; ob = circle ob o b p; oc = circle oc o c p; x = on_tline x ob c a, on_tline x oc a b; y = on_tline y oc a b, on_tline y oa b c; z = on_tline z oa b c, on_tline z ob c a; w = circle w x y z ? perp w p o p
imo_sl_2019_g1
a b c = triangle a b c; f = on_line f b c; g = on_line g b c; o = circle o a f g; d = intersection_lc d b o a; e = intersection_lc e c o a; u = circle u b d f; v = circle v c e g; t = on_tline t f u f, on_tline t g v g ? para a t b c
imo_sl_2019_g2
a b c = triangle a b c; d = foot d a b c; e = foot e b c a; f = foot f c a b; ib = incenter ib b d f; m = foot m ib d f; ic = incenter ic c d e; n = foot n ic d e; p = intersection_lc p n ib m; q = intersection_lc q m ic n ? cong m p n q
imo_sl_2019_g3
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 ? cyclic p q p1 q1
imo_sl_2019_g6
a b c = triangle a b c; d e f i = incenter2 d e f i a b c; o = circle o a b c; p = on_line p e f, on_circle p o a; q = intersection_lc q e o p; x = on_aline x p d a q d ? eqangle a p p x p i i q
imo_sl_2019_g7
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 ? perp a t a i
imo_sl_2020_g2
p a b = triangle p a b; z = angle_bisector z p b a, angle_bisector z p a b; z1 = on_aline z1 b a a b z, on_aline z1 a b b a z; d = on_aline d p a p b z1, on_aline d a p a b z; c = on_aline c p b p a z1, on_aline c b p b a z; q = angle_bisector q a d p, angle_bisector q p c b ? cong a q b q
imo_sl_2020_g3a
a b c = triangle a b c; d = eqangle2 d a b c; e = reflect e a b c; f = reflect f a c d; k = intersection_ll k a e b d; l = intersection_ll l a f b d; o = circle o b e k; w = circle w d f l; x = reflect x a b d ? cong o x o b
imo_sl_2020_g3b
a b c = triangle a b c; d = eqangle2 d a b c; e = reflect e a b c; f = reflect f a c d; k = intersection_ll k a e b d; l = intersection_ll l a f b d; o = circle o b e k; w = circle w d f l; x = reflect x a b d ? cong w x w d
imo_sl_2020_g3c
a b c = triangle a b c; d = eqangle2 d a b c; e = reflect e a b c; f = reflect f a c d; k = intersection_ll k a e b d; l = intersection_ll l a f b d; o = circle o b e k; w = circle w d f l; x = reflect x a b d ? coll o w x
imo_sl_2020_g6
a b c = triangle a b c; i = incenter i a b c; ia = excenter ia a b c; d = foot d i b c; e = intersection_ll e a d b ia; f = intersection_ll f a d c ia; o = circle o a i d; w = circle w ia e f; x = on_circle x o a, on_circle x w ia ? coll o x w
imo_sl_2020_g6a
a b c = triangle a b c; i = incenter i a b c; ia = excenter ia a b c; d = foot d i b c; e = intersection_ll e a d b ia; f = intersection_ll f a d c ia; o = circle o a i d; w = circle w ia e f; t = on_circle t w ia, on_circum t b c i ? cyclic t a i d
imo_sl_2020_g6b
a b c = triangle a b c; i = incenter i a b c; ia = excenter ia a b c; d = foot d i b c; e = intersection_ll e a d b ia; f = intersection_ll f a d c ia; o = circle o a i d; w = circle w ia e f; t = on_circle t w ia, on_circum t b c i ? cyclic t ia e f
imo_sl_2020_g6c
a b c = triangle a b c; i = incenter i a b c; ia = excenter ia a b c; d = foot d i b c; e = intersection_ll e a d b ia; f = intersection_ll f a d c ia; o = circle o a i d; w = circle w ia e f; t = on_circle t w ia, on_circum t b c i ? coll t o w
imo_sl_2020_g7a
a b c = triangle a b c; ma = midpoint ma b c; mb = midpoint mb c a; mc = midpoint mc a b; w = circle w a b c; p = on_circle p w a; d = reflect d p mb mc; e = reflect e p mc ma; f = reflect f p ma mb; oa = circle oa a d p; ob = circle ob b e p; oc = circle oc c f p; u = intersection_cc u oa ob p ? cong oc u oc c
imo_sl_2020_g7b
a b c = triangle a b c; ma = midpoint ma b c; mb = midpoint mb c a; mc = midpoint mc a b; w = circle w a b c; p = on_circle p w a; d = reflect d p mb mc; e = reflect e p mc ma; f = reflect f p ma mb; oa = circle oa a d p; ob = circle ob b e p; oc = circle oc c f p; u = intersection_cc u oa ob p; a1 = foot a1 a b c; b1 = foot b1 b c a; c1 = foot c1 c a b; h = intersection_ll h a a1 b b1; x = on_bline x b e, on_bline x c f; y = on_bline y a d, on_bline y c f; z = on_bline z a d, on_bline z b e; o = circle o x y z ? cong o u o x
imo_sl_2020_g8
a b c = triangle a b c; o = circle o a b c; i = incenter i a b c; ob = on_bline ob i b; oc = on_bline oc i c, on_line oc i ob; p = intersection_cc p ob o b; m = intersection_lc m a ob b; q = intersection_cc q oc o c; n = intersection_lc n a oc c; x = intersection_ll x p m q n; y = on_tline y b ob b, on_tline y c oc c ? coll a x y
imo_sl_2021_g1
c a b = iso_triangle c a b; d = parallelogram d a b c; p = on_line p a b; o = circle o a c d; q = on_line q p d, on_circle q o a; w = circle w a p q; r = on_line r p c, on_circle r w a; x = on_line x c d, on_line x a q ? coll x b r
imo_sl_2021_g4
a b c = triangle a b c; o = circle o a b c; d = on_circle d o a; e = on_tline e d o d, on_line e a b; f = on_tline f d o d, on_line f b c; t = on_pline t e c d, on_pline t f a d; k = on_line k d f, eqdistance k t t d; x = on_line x a c, on_line x d t ? coll x b k
imo_sl_2021_g7
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
imo_sl_2021_g8
a b c = triangle a b c; o = circle o a b c; u v w i = excenter2 u v w i a b c; x = on_circle x o a, on_circle x i u; y = intersection_cc y o i x; p = on_tline p x i x, on_pline p a i x; q = on_tline q y i y, on_pline q a i y; o1 = circle o1 a p x; o2 = circle o2 a q y; r = on_tline r p o1 p, on_tline r q o2 q ? perp a r b c
imo_sl_2022_g1
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
imo_sl_2022_g2
a b c = triangle a b c; f = foot f a b c; p = on_line p a f; d = on_pline d p a c, on_line d b c; e = on_pline e p a b, on_line e b c; o = circle o a b d; x = intersection_cc x o d a; w = circle w a c e; y = intersection_cc y w e a ? cyclic b c x y
imo_sl_2022_g3
a b c = triangle a b c; o = circle o a b c; d = on_circle d o a; w = on_tline w a a c, on_bline w a d; q = intersection_lc q b w a; x = on_tline x b b d, on_bline x b c; p = intersection_lc p a x b; m = midpoint m b c; n = midpoint n a d; u = circle u a n q; v = circle v b m p; z = on_tline z a u a, on_tline z b v b ? coll c d z
imo_sl_2022_g4
b c = segment b c; d = on_line d b c; o = on_bline o b c; w = on_tline w d b c, eqdistance w o d o; a = on_line a o w, on_circle a o b; x = intersection_ll x d w a c; y = intersection_ll y d w a b; u = circle u a x y; z = intersection_cc z o u a ? perp d z u z
imo_sl_2022_g5a
a b c = triangle a b c; x1 = on_line x1 b c; y1 = on_line y1 c a; z1 = intersection_ll z1 a b x1 y1; p1 = intersection_tt p1 y1 c a z1 a b; q1 = intersection_tt q1 x1 b c z1 a b; r1 = intersection_tt r1 x1 b c y1 c a; o1 = circle o1 p1 q1 r1; x2 = on_line x2 b c; y2 = intersection_lp y2 c a x2 x1 y1; z2 = intersection_lp z2 a b x2 x1 y1; p2 = intersection_tt p2 y2 c a z2 a b; q2 = intersection_tt q2 x2 b c z2 a b; r2 = intersection_tt r2 x2 b c y2 c a; o2 = circle o2 p2 q2 r2; h = intersection_ll h p1 p2 q1 q2 ? coll h o1 o2
imo_sl_2022_g5b
a b c = triangle a b c; x1 = on_line x1 b c; y1 = on_line y1 c a; z1 = intersection_ll z1 a b x1 y1; p1 = intersection_tt p1 y1 c a z1 a b; q1 = intersection_tt q1 x1 b c z1 a b; r1 = intersection_tt r1 x1 b c y1 c a; o1 = circle o1 p1 q1 r1; x2 = on_line x2 b c; y2 = intersection_lp y2 c a x2 x1 y1; z2 = intersection_lp z2 a b x2 x1 y1; p2 = intersection_tt p2 y2 c a z2 a b; q2 = intersection_tt q2 x2 b c z2 a b; r2 = intersection_tt r2 x2 b c y2 c a; o2 = circle o2 p2 q2 r2; h = intersection_ll h p1 p2 q1 q2 ? cong o1 p1 o1 h
imo_sl_2022_g5c
a b c = triangle a b c; x1 = on_line x1 b c; y1 = on_line y1 c a; z1 = intersection_ll z1 a b x1 y1; p1 = intersection_tt p1 y1 c a z1 a b; q1 = intersection_tt q1 x1 b c z1 a b; r1 = intersection_tt r1 x1 b c y1 c a; o1 = circle o1 p1 q1 r1; x2 = on_line x2 b c; y2 = intersection_lp y2 c a x2 x1 y1; z2 = intersection_lp z2 a b x2 x1 y1; p2 = intersection_tt p2 y2 c a z2 a b; q2 = intersection_tt q2 x2 b c z2 a b; r2 = intersection_tt r2 x2 b c y2 c a; o2 = circle o2 p2 q2 r2; h = intersection_ll h p1 p2 q1 q2 ? cong o2 p2 o2 h
imo_sl_2022_g6
a b c = triangle a b c; h = foot h a b c; r = on_line r a h; p = angle_mirror p c b r, angle_mirror p b c r; e = intersection_ll e a c b r; f = intersection_ll f a b c r; q = intersection_ll q e f a h; k = angle_mirror k b c a, angle_mirror k c b a ? coll p q k
imo_sl_2022_g7
a b c = triangle a b c; o = circle o a b c; h = orthocenter h a b c; a1 = on_circle a1 o a; d1 = intersection_lc d1 h o a1; b1 = on_bline b1 h d1, on_circle b1 o a; c1 = on_bline c1 h d1, on_circle c1 o a; a2 = on_circle a2 o a; x = intersection_ll x a a1 b b1; y = intersection_ll y b b1 c c1; z = intersection_ll z c c1 a a1; w = circle w x y z ? coll w o h
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
2009_sl_g3
a b c = triangle a b c; x y z i = incenter2 x y z i a b c; g = on_line g b y, on_line g c z; r = parallelogram r b c y; s = parallelogram s z b c ? cong g r g s
2009_sl_g6
a b c d = quadrangle a b c d; p = on_line p a d, on_line p b c; o1 = circle o1 a b p; o2 = circle o2 d c p; h1 = orthocenter h1 a b p; h2 = orthocenter h2 d c p; e1 = midpoint e1 o1 h1; e2 = midpoint e2 o2 h2; z = on_tline z e1 c d, on_tline z e2 a b ? coll z h1 h2
2010_sl_g1
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; o = circle o a b c; p = on_line p e f, on_circle p o a; q = on_line q b p, on_line q d f ? cong a p a q
2010_sl_g2
m l k = iso_triangle m l k; o = circle o m l k; p = free p; a = on_circle a o m, on_line a k p; b = on_circle b o m, on_line b l p; c = on_circle c o m, on_line c m p; s = on_line s a b, on_tline s c c o ? cong s c s p
2011_sl_g6
a b c = iso_triangle a b c; d = midpoint d a c; o = circle o d b c; e = on_circle e o d, angle_bisector e b a c; o2 = circle o2 a e b; f = on_line f b d, on_circle f o2 a; i = on_line i a f, on_line i b e; k = on_line k c i, on_line k b d; x = foot x i a b; y = foot y i b k; z = foot z i a k ? circle i x y z
2012_sl_g2
a b c = triangle a b c; o = circle o a b c; d = on_circle d o a; e = on_line e a c, on_line e b d; f = on_line f b c, on_line f a d; g = parallelogram g d e c; h = reflect h e a d ? cyclic d h f g
2012_sl_g3
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; i1 = incenter i1 a e f; i2 = incenter i2 b d f; o1 = circle o1 a c i1; o2 = circle o2 b c i2 ? para o1 o2 i1 i2
2012_sl_g4
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; m = midpoint m b c; e = mirror e d m; x = on_tline x d b c, on_line x a o; y = on_tline y e b c, on_line y a d ? cyclic b x c y
2013_sl_g2
a b c = triangle a b c; w = circle w a b c; m = midpoint m a b; n = midpoint n a c; t = on_bline t b c, on_circle t w a; w2 = circle w2 a m t; w3 = circle w3 a n t; x = on_circle x w2 a, on_bline x a c; y = on_circle y w3 a, on_bline y a b; k = on_line k m n, on_line k x y ? cong k a k t
2013_sl_g4
a b c = triangle a b c; p = on_line p a c, on_aline p b a b c a; q = on_line q a c, on_aline q b a a c b; d = on_line d b q, on_circle d p b; o = circle o a b c; r = on_circle r o a, on_line r a d ? cong q b q r
2014_sl_g3
a b c = triangle a b c; o = circle o a b c; m = angle_bisector m a b c, on_circle m o a; f = midpoint f b m; p = angle_bisector p a o b, on_circle p f b; q = angle_bisector q b o c, on_circle q f b; r = on_line r p q, on_bline r b m ? para a c b r
2015_sl_g1
a b c = triangle a b c; h = orthocenter h a b c; g = parallelogram g h a b; i1 = on_line i1 a c, on_line i1 h g; i = mirror i h i1; o = circle o g c i; j = on_circle j o g, on_line j a c ? cong i j a h
2015_sl_g3
c a b = r_triangle c a b; h = foot h c a b; z = on_line z c h; d = mirror d a z; p = on_line p b d, on_line p c h; o = midpoint o b d; q = on_dia q p o, on_circle q o b; y = on_line y a d, on_line y c q ? cong o y o b
2015_sl_g5
a b c = triangle a b c; d = midpoint d a b; f = midpoint f a c; g = midpoint g b c; o = on_bline o d c, on_tline o d a b; h = on_line h a f, on_circle h o d; i = on_line i b g, on_circle i o d; h1 = mirror h1 h f; i1 = mirror i1 i g; q = on_line q h1 i1, on_line q c d; m = on_line m h1 i1, on_line m f g; p = on_line p c m, on_circle p o d ? cong c q q p
2016_sl_g2
a b c = triangle a b c; i = incenter i a b c; w = circle w a b c; m = midpoint m b c; d = foot d i b c; f = on_line f a b, on_tline f i a i; e = on_line e a c, on_tline e i a i; o = circle o a e f; x = on_circle x o a, on_circle x w a; y = on_line y x d, on_line y a m ? cyclic y a b c
2016_sl_g4
a b c = iso_triangle a b c; i = incenter i a b c; d = on_line d b i, on_line d a c; e = on_tline e d a c, on_line e a i; z = reflect z i a c ? cyclic z b d e
2016_sl_g5
a b c = triangle a b c; o = circle o a b c; i = orthocenter i a b c; d = foot d a o i; x = on_line x a b; s = circle s x a d; y = on_line y a c, on_circle y s a; p = foot p a b c; m = midpoint m b c; w2 = circle w2 x s y ? cong w2 p w2 m
2016_sl_g6
d b c = triangle d b c; a = eqangle2 a b c d; e = on_line e a c, angle_bisector e a b c; f = on_line f a c, angle_bisector f a d c; p = on_line p e b, on_line p d f; m = midpoint m a c; o = circle o b p d; x = on_line x b m, on_circle x o b; y = on_line y d m, on_circle y o b; q = on_line q x e, on_line q y f ? perp a c p q
2017_sl_g3
a b c = triangle a b c; o = circle o a b c; z = foot z b c a; p = on_line p b z, on_line p o a; y = foot y c a b; q = on_line q o a, on_line q c y; h = on_line h b z, on_line h c y; o2 = circle o2 p q h; m = midpoint m b c ? coll o2 a m
2017_sl_g4
a b c = triangle a b c; d e f o = excenter2 d e f o a b c; o2 = circle o2 a e f; p = on_line p b c, on_circle p o2 a; q = on_line q b c, on_circle q o2 a; m = midpoint m a d; o3 = circle o3 m p q; z = on_circle z o3 m, on_circle z o d ? coll z o o3
2017_sl_g7
r s t = triangle r s t; i = circle i r s t; q = on_circle q i r; a = on_tline a r r i, on_tline a s s i; b = on_tline b s s i, on_tline b t t i; c = on_tline c t t i, on_tline c q q i; d = on_tline d q q i, on_tline d r r i; ia = incenter ia d a b; ib = incenter ib a b c; ic = incenter ic b c d; id = incenter id c d a; oa = circle oa a ib id; oc = circle oc ib id c; x1 x2 x3 x4 = cc_tangent x1 x2 x3 x4 oa a oc c; x = on_line x x1 x2, on_line x x3 x4; ob = circle ob b ia ic; od = circle od d ia ic; y1 y2 y3 y4 = cc_tangent y1 y2 y3 y4 ob b od d; y = on_line y y1 y2, on_line y y3 y4 ? perp x i i y
2018_sl_g4a
a b c = triangle a b c; t = free t; a1 = reflect a1 t b c; b1 = reflect b1 t a c; c1 = reflect c1 t a b; o = circle o a1 b1 c1; a2 = on_line a2 a1 t, on_circle a2 o a1; b2 = on_line b2 b1 t, on_circle b2 o a1; c2 = on_line c2 c1 t, on_circle c2 o a1; z = on_line z a a2, on_line z b b2 ? coll z c c2
2018_sl_g4b
a b c = triangle a b c; t = free t; a1 = reflect a1 t b c; b1 = reflect b1 t a c; c1 = reflect c1 t a b; o = circle o a1 b1 c1; a2 = on_line a2 a1 t, on_circle a2 o a1; b2 = on_line b2 b1 t, on_circle b2 o a1; c2 = on_line c2 c1 t, on_circle c2 o a1; z = on_line z a a2, on_line z b b2 ? cyclic z a1 b1 c1
2018_sl_g5
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; e = on_line e b i; f = on_line f c i, on_line f d e; x1 = on_bline x1 b e, on_bline x1 c f; y1 = on_bline y1 a d, on_bline y1 c f; z1 = on_bline z1 a d, on_bline z1 b e; o2 = circle o2 x1 y1 z1; t = on_circle t o2 x1, on_circle t o a ? coll t o2 o
2018_sl_g7
a b c = triangle a b c; o = circle o a b c; p = on_circle p o a; oa = circle oa a o p; ob = circle ob b o p; oc = circle oc c o p; la = on_tline la oa b c; lb = on_tline lb ob c a; lc = on_tline lc oc a b; z = on_line z la oa, on_line z lb ob; y = on_line y lb ob, on_line y lc oc; x = on_line x lc oc, on_line x la oa; o2 = circle o2 x y z ? perp o p o2 p
2019_sl_g1
a b c = triangle a b c; d = on_line d a b; e = on_line e a c; o = circle o a d e; f = on_line f b c, on_circle f o a; g = on_line g b c, on_circle g o a; o2 = circle o2 b d f; o3 = circle o3 c e g; t = on_tline t f f o2, on_tline t g g o3 ? para a t b c
2019_sl_g2
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; ob = incenter ob b d f; oc = incenter oc c d e; m = on_line m d f, on_tline m ob d f; n = on_line n d e, on_tline n oc d e; p = on_line p m n, on_circle p ob m; q = on_line q m n, on_circle q oc n ? cong m p n q
2019_sl_g7
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 a r, on_circle p i d; o2 = circle o2 p c e; o3 = circle o3 p b f; q = on_circle q o2 p, on_circle q o3 p; z = on_line z d i, on_line z p q ? eqangle b a a z a z a c
2020_sl_g7a
a b c = triangle a b c; o = circle o a b c; p = on_circle p o a; ma = midpoint ma b c; mb = midpoint mb a c; mc = midpoint mc a b; d = reflect d p mb mc; e = reflect e p ma mc; f = reflect f p ma mb; wa = circle wa a d p; wb = circle wb b e p; wc = circle wc c f p; x = on_bline x a d, on_bline x b e; y = on_bline y c f, on_bline y b e; z = on_bline z c f, on_bline z a d; w = circle w x y z; q = on_circle q wa a, on_circle q wb b ? cyclic q c f p
2020_sl_g7b
a b c = triangle a b c; o = circle o a b c; p = on_circle p o a; ma = midpoint ma b c; mb = midpoint mb a c; mc = midpoint mc a b; d = reflect d p mb mc; e = reflect e p ma mc; f = reflect f p ma mb; wa = circle wa a d p; wb = circle wb b e p; wc = circle wc c f p; x = on_bline x a d, on_bline x b e; y = on_bline y c f, on_bline y b e; z = on_bline z c f, on_bline z a d; q = on_circle q wa a, on_circle q wb b ? cyclic q x y z
2020_sl_g8
a b c = triangle a b c; i = incenter i a b c; o = circle o a b c; wb = on_bline wb b i; wc = on_bline wc c i, on_line wc i wb; p = on_circle p wb i, on_circle p o a; m = on_circle m wb i, on_line m a b; q = on_circle q wc i, on_circle q o a; n = on_circle n wc i, on_line n a c; x = on_line x p m, on_line x q n; y = on_tline y b b wb, on_tline y c c wc ? coll a x y
2021_sl_g1
c a b = iso_triangle c a b; d = parallelogram d a b c; p = on_line p a b; o = circle o a c d; q = on_line q p d, on_circle q o a; o2 = circle o2 a p q; r = on_line r p c, on_circle r o2 a; y = on_line y c d, on_line y a q ? coll y b r
2021_sl_g4
a b c = triangle a b c; o = circle o a b c; d = on_circle d o a; e = on_tline e d d o, on_line e b a; f = on_tline f d d o, on_line f b c; t = on_pline t e c d, on_pline t f a d; k = on_line k d f, on_circle k t d; z = on_line z a c, on_line z d t ? coll z b k
2022_sl_g2
a b c = triangle a b c; f = foot f a b c; p = on_line p a f; d = on_pline d p a c, on_line d b c; e = on_pline e p a b, on_line e b c; o1 = circle o1 a b d; o2 = circle o2 a c e; x = on_circle x o1 a, on_circle x d a; y = on_circle y o2 a, on_circle y e a ? cyclic b c x y
2022_sl_g3
a b c = triangle a b c; d = on_circum d a b c; o1 = on_tline o1 a a c, on_bline o1 a d; o2 = on_tline o2 b b d, on_bline o2 b c; q = on_circle q o1 a, on_line q a b; p = on_circle p o2 b, on_line p a b; m = midpoint m b c; n = midpoint n a d; w1 = circle w1 a n q; x = on_line x c d, on_tline x a a w1; w2 = circle w2 b m p ? perp x b b w2
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