Exsymmedian

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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 $Ea, Eb, Ec$ being the symmedians through the vertices $sa, sb, sc$, two exsymmedians and one symmedian intersect in a common point:

$$\begin{align} E_a&=e_b \cap e_c \cap s_a \\ E_b&=e_a \cap e_c \cap s_b \\ E_c&=e_a \cap e_b \cap s_c \end{align} $$

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:

$$\begin{align} k_a&=a\cdot \frac{2\triangle}{c^2+b^2-a^2} \\[6pt] k_b&=b\cdot \frac{2\triangle}{c^2+a^2-b^2} \\[6pt] k_c&=c\cdot \frac{2\triangle}{a^2+b^2-c^2} \end{align} $$

Here $△ABC$ denotes the area of the triangle $△ABC$, and $ea, eb, ec$ denote the perpendicular line segments connecting the triangle sides $sa, sb, sc$ with the exsymmedian points $A, B, C$.