Clawson point

In Euclidean geometry, the Clawson point is a special point in a triangle defined by the trilinear coordinates tan α : tan β : tan γ,[1] where α, β, γ are the interior angles at the triangle vertices A, B, C. It is named after John Wentworth Clawson, who published it 1925 in the American Mathematical Monthly.

It is denoted X(19) in Clark Kimberling's Encyclopedia of Triangle Centers.

Hence the three lines AA', BB', CC' meet in the Clawson point.

[4] However the French mathematician Émile Lemoine had already examined the point in 1886.

[5] Later the point was independently rediscovered by R. Lyness and G. R. Veldkamp in 1983, who called it crucial point after the Canadian math journal Crux Mathematicorum in which it was published as problem 682.

Construction 1: Clawson point as a homothetic center
Reference triangle △ ABC
Extended sides of △ ABC
Excircles of △ ABC
Orthic triangle △ H A H B H C , and triangle △ T A T B T C , internally tangent to each pair of excircles
Perspective lines between △ H A H B H C and △ T A T B T C , which are similar ; meet at the homothetic center P (the Clawson point )
Construction 2: Clawson point as a center of perspective
Reference triangle △ ABC
Extended sides of △ ABC
Circumcircle and excircles of △ ABC
Triangle △ A'B'C', formed by lines connecting pairs of intersection points between each excircle and the circumcircle
Perspective lines between △ ABC and △ A'B'C', meeting at the perspective center P (the Clawson point )