Congruent isoscelizers point

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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

P 1 Q 1 ¯ = P 2 Q 2 ¯ = P 3 Q 3 ¯ {\displaystyle {\overline {P_{1}Q_{1}}}={\overline {P_{2}Q_{2}}}={\overline {P_{3}Q_{3}}}} {\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"

cos ⁡ B 2 + cos ⁡ C 2 − cos ⁡ A 2 : cos ⁡ C 2 + cos ⁡ A 2 − cos ⁡ B 2 : cos ⁡ A 2 + cos ⁡ B 2 − cos ⁡ C 2 = tan ⁡ A 2 + sec ⁡ A 2     : tan ⁡ B 2 + sec ⁡ B 2 : tan ⁡ C 2 + sec ⁡ C 2 {\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}}} {\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

References

  1. Kimberling, Clark. "X(173) = Congruent isoscelizers point". Encyclopedia of Triangle Centers. Archived from the original on 19 April 2012. Retrieved 3 June 2012.
  2. Kimberling, Clark. "Congruent isoscelizers point". Retrieved 3 June 2012.