Exsymmedian

Triangle exsymmedians (red): Symmedians (green): Exsymmedian points (red): ${\displaystyle ABC}$
${\displaystyle e_{a},e_{b},e_{c}}$
${\displaystyle s_{a},s_{b},s_{c}}$
${\displaystyle E_{a},E_{b},E_{c}}$

The exsymmedians of a triangle are defined as the tangents to the perimeter of the triangle at the corner points of the triangle. These three exsymmedians form a new triangle, the corner points of which are called exsymmedian points.

A symmedian each also runs through the exsymmedian points , that is to say two exsymmedians and an associated symmedian intersect at a common point. More precisely, for a triangle with exsymmedians , symmedians and exsymmedian points : ${\displaystyle ABC}$ ${\displaystyle e_{a},e_{b},e_{c}}$ ${\displaystyle s_{a},s_{b},s_{c}}$ ${\displaystyle E_{a},E_{b},E_{c}}$

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

The length of the vertical connection between an exsymmedian point and the associated triangle side is proportional to this triangle side and the following formulas apply:

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

In this case denote the area of the triangle and the vertical lines connecting the sides of the triangle and the Exsymmedian points . ${\displaystyle \triangle }$ ${\displaystyle ABC}$ ${\displaystyle k_{a},k_{b},k_{c}}$ ${\displaystyle a,b,c}$ ${\displaystyle E_{a},E_{b},E_{c}}$

literature

• Roger A. Johnson: Advanced Euclidean Geometry . Dover 2007, ISBN 978-0-486-46237-0 , pp. 214-215 (first published in 1929 by the Houghton Mifflin Company (Boston) under the title Modern Geometry ).