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

{\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 $P 1$ and $Q 1$ , where $P 1$ lies on $AB$ and $Q 1$ on $AC$ , such that the triangle $△ AP 1 Q 1$ 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 $P 1 Q 1, P 2 Q 2, P 3 Q 3$ 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 $P 1 Q 1, P 2 Q 2, P 3 Q 3$ 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]

{\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]

References

1 2 3 4 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.

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[ETC-1] 1 2 3 4 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.

[1]

[2]

Definition

Properties

See also

References