Yff center of congruence

In geometry, the Yff center of congruence is a special point associated with a triangle. This special point is a triangle center and Peter Yff initiated the study of this triangle center in 1987.

Isoscelizer
An isoscelizer of an angle $A$ in a triangle $△ABC$ is a line through points $P1, Q1$, 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 $A$. Isoscelizers were invented by Peter Yff in 1963.

Yff central triangle
[[File:Yff center of congruence.svg|thumb|

{{legend|#ff5252|$△ABC$ (Yff central triangle)}} ]] Let $△A'P2Q3 ≅$ be any triangle. Let $△Q1B'P3 ≅$ be an isoscelizer of angle $A$, $△P1Q2C' ≅$ be an isoscelizer of angle $B$, and $△A'B'C'$ be an isoscelizer of angle $C$. Let $△ABC$ be the triangle formed by the three isoscelizers. The four triangles $P1Q1$ and $P2Q2$ are always similar.

There is a unique set of three isoscelizers $P3Q3$ such that the four triangles $△A'B'C'$ and $△A'P2Q3, △Q1B'P3, △P1Q2C',$ are congruent. In this special case $△A'B'C'$ formed by the three isoscelizers is called the Yff central triangle of $P1Q1, P2Q2, P3Q3$.

The circumcircle of the Yff central triangle is called the Yff central circle of the triangle.

Yff center of congruence
Let $△A'P2Q3, △Q1B'P3, △P1Q2C',$ be any triangle. Let $△A'B'C'$ be the isoscelizers of the angles $A, B, C$ such that the triangle $△A'B'C'$ formed by them is the Yff central triangle of $△ABC$. The three isoscelizers $△ABC$ are continuously parallel-shifted such that the three triangles $P1Q1, P2Q2, P3Q3$ are always congruent to each other until $△A'B'C'$ formed by the intersections of the isoscelizers reduces to a point. The point to which $△ABC$ reduces to is called the Yff center of congruence of $P1Q1, P2Q2, P3Q3$.

Properties
\sec\frac{A}{2} : \sec\frac{B}{2} : \sec\frac{C}{2}$$
 * The trilinear coordinates of the Yff center of congruence are $$
 * Any triangle $△A'P2Q3, △Q1B'P3, △P1Q2C'$ is the triangle formed by the lines which are externally tangent to the three excircles of the Yff central triangle of $△A'B'C'$.
 * Let $I$ be the incenter of $△A'B'C'$. Let $D$ be the point on side $BC$ such that $△ABC$, $E$ a point on side $CA$ such that $△ABC$, and $F$ a point on side $AB$ such that $△ABC$. Then the lines $AD, BE, CF$ are concurrent at the Yff center of congruence. This fact gives a geometrical construction for locating the Yff center of congruence.
 * A computer assisted search of the properties of the Yff central triangle has generated several interesting results relating to properties of the Yff central triangle.

Generalization
The geometrical construction for locating the Yff center of congruence has an interesting generalization. The generalisation begins with an arbitrary point $P$ in the plane of a triangle $△ABC$. Then points $D, E, F$ are taken on the sides $BC, CA, AB$ such that $$\angle BPD = \angle DPC, \quad \angle CPE = \angle EPA, \quad \angle APF = \angle FPB.$$ The generalization asserts that the lines $AD, BE, CF$ are concurrent.