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

Through any point P draw the line parallel to BC as well as the corresponding antiparallel. Then the A-isoscelizer through P 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 u_1=(u_(1x),u_(1y)) and u_2=(u_(2x),u_(2y)) be the unit vectors from a given vertex v=(v_x,v_y), let X=(x,y) be a point in the interior of a triangle through which an isoscelizer passes, and the side lengths of the isosceles triangle be l. Then setting the point-line distance from the vector (u_1,u_2) to the point x equal to 0 gives


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.

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