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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''.
*The
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
the congruent isoscelizers point is a special point associated with a plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
. It is a triangle center and it is listed as X(173) in Clark Kimberling
Clark Kimberling (born November 7, 1942 in Hinsdale, Illinois) is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer seq ...
's Encyclopedia of Triangle Centers
The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.
, the l ...
. This point was introduced to the study of triangle geometry by Peter Yff in 1989.
Definition
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''.
Properties
*The trilinear coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...
of the congruent isoscelizers point of triangle ''ABC'' are
::( 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 ) )
*The intouch triangle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
of the intouch triangle of triangle ''ABC'' is perspective to triangle ''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 triangle ''ABC''.
See also
*Yff center of congruence
In geometry, the Yff center of congruence is a special point associated with a triangle. This special point is a triangle center and Peter Yff initiated the study of this triangle center in 1987.
Isoscelizer
An isoscelizer of an angle ''A'' in ...
* Equal parallelians point
References
{{reflist Triangle centers