The exsymmedians are three lines associated with a

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...

. More precisely for a given triangle the exsymmedians are the tangent lines on the triangle's

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every pol ...

through the three vertices of the triangle. The triangle formed by the three exsymmedians is the

tangential triangle In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertices. Thus the incircle of the ...

and 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

symmedian In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the cor ...

s through the vertices

A, B, C

two exsymmedians and one symmedian intersect in a common point, that is: :

\begin
 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

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: :

k_c&=c\cdot \frac \end

Here

\triangle

denotes the area of the triangle

ABC

and

k_a, k_b, k_c

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''). Straight lines defined for a triangle