Congruent isoscelizers point

In geometry, the congruent isoscelizers point is a special point associated with a plane triangle.

It is a triangle center and it is listed as X(173) in Clark Kimberling's Encyclopedia of Triangle Centers.

This point was introduced to the study of triangle geometry by Peter Yff in 1989.

[1][2] An isoscelizer of an angle A in a triangle △ABC is a line through points P1 and Q1, where P1 lies on AB and Q1 on AC, such that the triangle △AP1Q1 is an isosceles triangle.

An isoscelizer of angle A is a line perpendicular to the bisector of angle A.

Let P1Q1, P2Q2, P3Q3 be the isoscelizers of the angles A, B, C respectively such that they all have the same length.

Then, for a unique configuration, the three isoscelizers P1Q1, P2Q2, P3Q3 are concurrent.

The point of concurrence is the congruent isoscelizers point of triangle △ABC.

{\displaystyle {\begin{array}{ccccc}\cos {\frac {B}{2}}+\cos {\frac {C}{2}}-\cos {\frac {A}{2}}&:&\cos {\frac {C}{2}}+\cos {\frac {A}{2}}-\cos {\frac {B}{2}}&:&\cos {\frac {A}{2}}+\cos {\frac {B}{2}}-\cos {\frac {C}{2}}\\[4pt]=\quad \tan {\frac {A}{2}}+\sec {\frac {A}{2}}\quad \ \ &:&\tan {\frac {B}{2}}+\sec {\frac {B}{2}}&:&\tan {\frac {C}{2}}+\sec {\frac {C}{2}}\end{array}}}