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.^[1][2]

Definition

${\displaystyle {\overline {P_{1}Q_{1}}}={\overline {P_{2}Q_{2}}}={\overline {P_{3}Q_{3}}}}$

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.^[1]

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^[1]

$${\displaystyle {\begin{array}{ccccc}\cos {\frac {B}{2}}+\cos {\frac {C}{2}}-\cos {\frac {A}{2}}&:&\cos {\frac {C}{2}}+\cos {\frac {A}{2}}-\cos {\frac {B}{2}}&:&\cos {\frac {A}{2}}+\cos {\frac {B}{2}}-\cos {\frac {C}{2}}\\[4pt]=\quad \tan {\frac {A}{2}}+\sec {\frac {A}{2}}\quad \ \ &:&\tan {\frac {B}{2}}+\sec {\frac {B}{2}}&:&\tan {\frac {C}{2}}+\sec {\frac {C}{2}}\end{array}}}$$

• 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.^[1]

See also

• Yff center of congruence
• Equal parallelians point