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.

Definition
An isoscelizer of an angle $A$ in a triangle $△ABC$ is a line through points $P_{1}$ and $Q_{1}$, where $P_{1}$ lies on $AB$ and $Q_{1}$ on $AC$, such that the triangle $△AP_{1}Q_{1}$ 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 $P_{1}Q_{1}, P_{2}Q_{2}, P_{3}Q_{3}$ 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 $P_{1}Q_{1}, P_{2}Q_{2}, P_{3}Q_{3}$ are concurrent. The point of concurrence is the congruent isoscelizers point of triangle $△ABC$.

Properties
[[File:Construction for congruent isoscelizers point.svg|thumb|300px|Construction for congruent isoscelizers point.

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 * The trilinear coordinates of the congruent isoscelizers point of triangle $△ABC$ are

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


 * 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 $△A'B'C'$.