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)
|
![l(u_(2y)-u_(1y))[(x-v_x)-lu_(1x)]-l(u_(2x)-u_(1x))[(y-v_y)-lu_(1y)]=0](/images/equations/Isoscelizer/NumberedEquation2.svg) |
(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.
