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 △ABC be any triangle. 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.[1]
The intouch triangle of the intouch triangle of triangle △ABC is perspective to △ABC, and the congruent isoscelizers point is the perspector. This fact can be used to locate by geometrical constructions the congruent isoscelizers point of any given △ABC.[1]