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

$\overline{P_1{Q}_1}=\overline{P_2{Q}_2}=\overline{P_3{Q}_3}$

An isoscelizer of an angle Template:Mvar in a triangle Template:Math is a line through points Template:Math and 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.

Let Template:Math be any triangle. Let Template:Math be the isoscelizers of the angles Template:Mvar respectively such that they all have the same length. Then, for a unique configuration, the three isoscelizers Template:Math are concurrent. The point of concurrence is the congruent isoscelizers point of triangle Template:Math.

Properties

Construction for congruent isoscelizers point. Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line

• The trilinear coordinates of the congruent isoscelizers point of triangle Template:Math are

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

• The intouch triangle of the intouch triangle of triangle Template:Math is perspective to Template:Math, 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 Template:Math.

See also

• Yff center of congruence
• Equal parallelians point