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 $e a, e b, e c$ being the exsymmedians and $s a, s b, s c$ being the symmedians through the vertices $A, B, C$ , two exsymmedians and one symmedian intersect in a common point:

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

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:

${\begin{aligned}k_{a}&=a\cdot {\frac {2\triangle }{c^{2}+b^{2}-a^{2}}}\\[6pt]k_{b}&=b\cdot {\frac {2\triangle }{c^{2}+a^{2}-b^{2}}}\\[6pt]k_{c}&=c\cdot {\frac {2\triangle }{a^{2}+b^{2}-c^{2}}}\end{aligned}}$

Here $△$ denotes the area of the triangle $△ ABC$ , and $k a, k b, k c$ denote the perpendicular line segments connecting the triangle sides $a, b, c$ with the exsymmedian points $E a, E b, E c$ .

References

Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0, pp. 214–215 (originally published 1929 with Houghton Mifflin Company (Boston) as Modern Geometry).