Exsymmedian

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 and its vertices, that is the three intersections of the exsymmedians are called exsymmedian points.

triangle ${\displaystyle ABC}$
exsymmedians (red): ${\displaystyle e_{a},e_{b},e_{c}}$
symmedians (green): ${\displaystyle s_{a},s_{b},s_{c}}$
exsymmedian points (red): ${\displaystyle E_{a},E_{b},E_{c}}$

For a triangle ${\displaystyle ABC}$ with ${\displaystyle e_{a},e_{b},e_{c}}$ being the exsymmedians and ${\displaystyle s_{a},s_{b},s_{c}}$ being the symmedians through the vertices ${\displaystyle A,B,C}$ two exsymmedians and one symmedian intersect in a common point, that is:

${\displaystyle {\begin{aligned}E_{a}&=e_{b}\cap e_{c}\cap s_{a}\\E_{b}&=e_{a}\cap e_{c}\cap s_{b}\\E_{c}&=e_{a}\cap e_{b}\cap s_{c}\end{aligned}}}$

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 ${\displaystyle \triangle }$ denotes the area of the triangle ${\displaystyle ABC}$ and ${\displaystyle k_{a},k_{b},k_{c}}$ the perpendicular line segments connecting the triangle sides ${\displaystyle a,b,c}$ with the exsymmedian points ${\displaystyle E_{a},E_{b},E_{c}}$.