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 Template:Math with Template:Mvar being the exsymmedians and Template:Mvar being the symmedians through the vertices Template:Mvar, two exsymmedians and one symmedian intersect in a common point:

$$\begin{array}{cc}{E}_a& =e_b\cap e_c\cap s_a\\ E_b& =e_a\cap e_c\cap s_b\\ E_c& =e_a\cap e_b\cap s_c\end{array}$$

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:

$$\begin{array}{cc}{k}_a& =a\cdot \frac{2\mathrm{△}}{c^2+b^2-a^2}\\ k_b& =b\cdot \frac{2\mathrm{△}}{c^2+a^2-b^2}\\ k_c& =c\cdot \frac{2\mathrm{△}}{a^2+b^2-c^2}\end{array}$$

Here Template:Math denotes the area of the triangle Template:Math, and Template:Mvar denote the perpendicular line segments connecting the triangle sides Template:Mvar with the exsymmedian points Template:Mvar.

References

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