Exsymmedian
In Euclidean geometry, the exsymmedians are three lines associated with a triangle.
More precisely, for a given triangle the exsymmedians are the tangent lines on the triangle's circumcircle through the three vertices of the triangle.
The triangle formed by the three exsymmedians is the tangential triangle; its vertices, that is the three intersections of the exsymmedians, are called exsymmedian points.
For a triangle △ABC with ea, eb, ec being the exsymmedians and sa, sb, sc being the symmedians through the vertices A, B, C, two exsymmedians and one symmedian intersect in a common point:
a
=
e
b
∩
e
{\displaystyle {\begin{aligned}E_{a}&=e_{b}\cap e_{c}\cap s_{a}\\E_{b}&=e_{a}\cap e_{c}\cap s_{b}\\E_{c}&=e_{a}\cap e_{b}\cap s_{c}\end{aligned}}}
The length of the perpendicular line segment connecting a triangle side with its associated exsymmedian point is proportional to that triangle side.
Specifically the following formulas apply:
{\displaystyle {\begin{aligned}k_{a}&=a\cdot {\frac {2\triangle }{c^{2}+b^{2}-a^{2}}}\\[6pt]k_{b}&=b\cdot {\frac {2\triangle }{c^{2}+a^{2}-b^{2}}}\\[6pt]k_{c}&=c\cdot {\frac {2\triangle }{a^{2}+b^{2}-c^{2}}}\end{aligned}}}
Here △ denotes the area of the triangle △ABC, and ka, kb, kc denote the perpendicular line segments connecting the triangle sides a, b, c with the exsymmedian points Ea, Eb, Ec.
