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# Isoscelizer

An isoscelizer of an (interior) angle in a triangle is a line through points where lies on and on such that is an isosceles triangle. An isoscelizer is therefore a line perpendicular to an angle bisector, and if the angle is , the line is known as an -isoscelizer. There are obviously an infinite number of isoscelizers for any given angle. Isoscelizers were invented by P. Yff in 1963.

Through any point draw the line parallel to as well as the corresponding antiparallel. Then the -isoscelizer through bisects the angle formed by the parallel and the antiparallel. Another way of saying this is that an isoscelizer is a line which is both parallel and antiparallel to itself.

Let and be the unit vectors from a given vertex , let be a point in the interior of a triangle through which an isoscelizer passes, and the side lengths of the isosceles triangle be . Then setting the point-line distance from the vector to the point equal to 0 gives

 (1)
 (2)
 (3)

Concatenation of six isoscelizers leads to a closed hexagon. The six vertices of this hexagon lie on a circle concentric with the incircle.

## See also

Angle Bisector, Antiparallel, Congruent Isoscelizers Point, Equal Parallelians Point, Isosceles Triangle, Yff Center of Congruence, Yff Central Triangle

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## Cite this as:

Weisstein, Eric W. "Isoscelizer." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Isoscelizer.html