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.^[1]

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.^[2]

Let △ABC be any triangle. Let P₁Q₁ be an isoscelizer of angle A, P₂Q₂ be an isoscelizer of angle B, and P₃Q₃ be an isoscelizer of angle C. Let △A'B'C' be the triangle formed by the three isoscelizers. The four triangles △A'P₂Q₃, △Q₁B'P₃, △P₁Q₂C', and △A'B'C' are always similar.

There is a unique set of three isoscelizers P₁Q₁, P₂Q₂, P₃Q₃ such that the four triangles △A'P₂Q₃, △Q₁B'P₃, △P₁Q₂C', and △A'B'C' are congruent. In this special case △A'B'C' formed by the three isoscelizers is called the Yff central triangle of △ABC.^[3]

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

Let △ABC be any triangle. Let P₁Q₁, P₂Q₂, P₃Q₃ 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 P₁Q₁, P₂Q₂, P₃Q₃ are continuously parallel-shifted such that the three triangles △A'P₂Q₃, △Q₁B'P₃, △P₁Q₂C' 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 △A'B'C' reduces to is called the Yff center of congruence of △ABC.

• The trilinear coordinates of the Yff center of congruence are^[1] $${\displaystyle \sec {\frac {A}{2}}:\sec {\frac {B}{2}}:\sec {\frac {C}{2}}}$$
• Any triangle △ABC is the triangle formed by the lines which are externally tangent to the three excircles of the Yff central triangle of △ABC.
• Let I be the incenter of △ABC. Let D be the point on side BC such that ∠BID = ∠DIC, E a point on side CA such that ∠CIE = ∠EIA, and F a point on side AB such that ∠AIF = ∠FIB. 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.^[4]
• 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.^[5]

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 $${\displaystyle \angle BPD=\angle DPC,\quad \angle CPE=\angle EPA,\quad \angle APF=\angle FPB.}$$ The generalization asserts that the lines AD, BE, CF are concurrent.^[4]

• Congruent isoscelizers point
• Central triangle