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Symmedial Triangle

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SymmedialTriangle

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

[0 b c; a 0 c; a b 0].
(1)

It is by definition perspective with the reference triangle, with perspector given by the symmedian point K. It is the cyclocevian triangle with respect to Kimberling center X_(1031).

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

It has area

Delta^'=(2a^2b^2c^2)/((a^2+b^2)(a^2+c^2)(b^2+c^2))Delta,
(2)

where Delta 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

a^'=(abcsqrt(a^4+a^2b^2-b^4+a^2c^2+3b^2c^2-c^4))/((a^2+b^2)(a^2+c^2))
(3)
b^'=(abcsqrt(-a^4+a^2b^2+b^4+3a^2c^2+b^2c^2-c^4))/((a^2+b^2)(b^2+c^2))
(4)
c^'=(abcsqrt(-a^4+3a^2b^2-b^4+a^2c^2+b^2c^2+c^4))/((a^2+c^2)(b^2+c^2)).
(5)

The symmedial circle is the circumcircle of the symmedial triangle.

See also

Brocard Inellipse, Symmedial Circle, Symmedian, Symmedian Point

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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

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