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 Template:Mvar in a triangle Template:Math is a line through points Template:Math, where Template:Math lies on Template:Mvar and Template:Math on Template:Mvar, such that the triangle Template:Math is an isosceles triangle. An isoscelizer of angle Template:Mvar is a line perpendicular to the bisector of angle Template:Mvar. Isoscelizers were invented by Peter Yff in 1963.

Yff central triangle

Template:Legend-line Template:Legend inline Template:Legend inline Template:Legend inline Template:Legend

Let Template:Math be any triangle. Let Template:Math be an isoscelizer of angle Template:Mvar, Template:Math be an isoscelizer of angle Template:Mvar, and Template:Math be an isoscelizer of angle Template:Mvar. Let Template:Math be the triangle formed by the three isoscelizers. The four triangles Template:Math and Template:Math are always similar.

There is a unique set of three isoscelizers Template:Math such that the four triangles Template:Math and Template:Math are congruent. In this special case Template:Math formed by the three isoscelizers is called the Yff central triangle of Template:Math.

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

Yff center of congruence

Animation showing the continuous shrinking of the Yff central triangle to the Yff center of congruence. The animation also shows the continuous expansion of the Yff central triangle until the three outer triangles reduce to points on the sides of the triangle.

Let Template:Math be any triangle. Let Template:Math be the isoscelizers of the angles Template:Mvar such that the triangle Template:Math formed by them is the Yff central triangle of Template:Math. The three isoscelizers Template:Math are continuously parallel-shifted such that the three triangles Template:Math are always congruent to each other until Template:Math formed by the intersections of the isoscelizers reduces to a point. The point to which Template:Math reduces to is called the Yff center of congruence of Template:Math.

Properties

Any triangle Template:Math is the triangle formed by the lines which are externally tangent to the three excircles of the Yff central triangle of Template:Math.

• The trilinear coordinates of the Yff center of congruence are $$\sec \frac{A}{2}:\sec \frac{B}{2}:\sec \frac{C}{2}$$
• Any triangle Template:Math is the triangle formed by the lines which are externally tangent to the three excircles of the Yff central triangle of Template:Math.
• Let Template:Mvar be the incenter of Template:Math. Let Template:Mvar be the point on side Template:Mvar such that Template:Math, Template:Mvar a point on side Template:Mvar such that Template:Math, and Template:Mvar a point on side Template:Mvar such that Template:Math. Then the lines Template:Mvar 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 of Yff centre of congruence

Generalization

The geometrical construction for locating the Yff center of congruence has an interesting generalization. The generalisation begins with an arbitrary point Template:Mvar in the plane of a triangle Template:Math. Then points Template:Mvar are taken on the sides Template:Mvar such that $$\mathrm{\angle }BPD=\mathrm{\angle }DPC,\mathrm{\angle }CPE=\mathrm{\angle }EPA,\mathrm{\angle }APF=\mathrm{\angle }FPB.$$ The generalization asserts that the lines Template:Mvar are concurrent.

See also

• Congruent isoscelizers point
• Central triangle