The symmedial triangle (a term coined here for the first time), is the
triangle whose vertices are the intersection points of the symmedians with the reference triangle . It has the very simple trilinear
vertex matrix

(1)

It is by definition perspective with the reference triangle , with perspector
given by the symmedian point . It is the cyclocevian
triangle with respect to Kimberling center .

The symmedial triangle is the polar triangle of
the Brocard inellipse .

It has area

(2)

where
is the area of the reference triangle (apparently
given incorrectly by Casey 1988, p. 172). This is the same area as the first
and second Brocard Cevian triangles .

It has side lengths

The symmedial circle is the circumcircle
of the symmedial triangle.

See also Brocard Inellipse ,

Symmedial Circle ,

Symmedian ,

Symmedian
Point
Explore with Wolfram|Alpha
References Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction
to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges,
Figgis, & Co., 1888. Referenced on Wolfram|Alpha Symmedial Triangle
Cite this as:
Weisstein, Eric W. "Symmedial Triangle."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/SymmedialTriangle.html

Subject classifications