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.

  Reference triangle ABC
  Circumcircle of ABC
  Exsymmedians (ea, eb, ec); intersect at the exsymmedian points (Ea, Eb, Ec)
  Symmedians (sa, sb, sc)

For a triangle ABC with ea, eb, ec being the exsymmedians and sa, sb, sc being the symmedians through the vertices A, B, C, two exsymmedians and one symmedian intersect in a common point:

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:

Here denotes the area of the triangle ABC, and ka, kb, kc denote the perpendicular line segments connecting the triangle sides a, b, c with the exsymmedian points Ea, Eb, Ec.

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).
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