热尔岗点存在定理
a b c = triangle a b c; i = incenter i a b c; d = foot d i b c; e = foot e i c a; f = foot f i a b; p = on_line p a d, on_line p b e ? coll p c f

Droz_farny定理
a b c = triangle a b c; h = orthocenter h a b c; t = on_line t a b; e = on_line e b c, on_line e t h; g = on_line g a c, on_line g t h; f = on_line f a c, on_tline f h t h; d = on_line d b c, on_tline d h t h; i = on_line i a b, on_tline i h t h; j = midpoint j i t; k = midpoint k d e; l = midpoint l f g ? coll j k l

帕斯卡定理
a b c = triangle a b c; o = circumcenter o a b c; d = on_circle d o a; e = on_circle e o a; f = on_circle f o a; p = on_line p a b, on_line p d e; q = on_line q b c, on_line q e f; r = on_line r c d, on_line r f a ? coll p q r