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 A in a triangle △ABC is a line through points P₁ and 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.

Let △ABC be any triangle. Let P₁Q₁, P₂Q₂, P₃Q₃ 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 P₁Q₁, P₂Q₂, P₃Q₃ are concurrent. The point of concurrence is the congruent isoscelizers point of triangle △ABC.

Properties

Construction for congruent isoscelizers point.

  Reference triangle △ABC

  Congruent isoscelizers of △ABC

  Incircle of △ABC

  Intouch triangle △A'B'C' of △ABC

  Incircle of △A'B'C' (used to construct △A"B"C")

  Perspective lines between △ABC and △A"B"C"

• 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 △ABC.

See also

• Yff center of congruence
• Equal parallelians point