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
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
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2d12bf752a6af1f2ceafd905105c07a7d64e6e)
- The perspector. This fact can be used to locate by geometrical constructions the congruent isoscelizers point of any given △ABC.[1]
See also
References
- ^ a b c d Kimberling, Clark. "X(173) = Congruent isoscelizers point". Encyclopedia of Triangle Centers. Archived from the original on 19 April 2012. Retrieved 3 June 2012.
- ^ Kimberling, Clark. "Congruent isoscelizers point". Retrieved 3 June 2012.