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Mixtilinear Triangle

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MixtilinearTriangle

The mixtilinear triangle is the triangle connecting the centers of the mixtilinear incircles.

It has trilinear vertex matrix

[1/2(cosA-cosB-cosC+1) 1 1; 1 1/2(-cosA+cosB-cosC+1) 1; 1 1 1/2(-cosA-cosB+cosC+1)].
(1)

In has area

Delta^'=(a^3-a^2b-ab^2+b^3-a^2c+6abc-b^2c-ac^2-bc^2+c^3)/((a+b+c)^3)Delta,
(2)

where Delta is the area of the reference triangle, and side lengths

a^'=(a(-a+b+c))/(a+b+c)sqrt((a^3+a^2b-5ab^2+3b^3+a^2c+10abc-3b^2c-5ac^2-3bc^2+3c^3)/((a+b-c)(a-b+c)(a+b+c)))
(3)
b^'=(b(a-b+c))/(a+b+c)sqrt((3a^3-5a^2b+ab^2+b^3-3a^2c+10abc+b^2c-3ac^2-5bc^2+3c^3)/((a+b-c)(-a+b+c)(a+b+c)))
(4)
c^'=((a+b-c)c)/(a+b+c)sqrt((3a^3-3a^2b-3ab^2+3b^3-5a^2c+10abc-5b^2c+ac^2+bc^2+c^3)/((a-b+c)(-a+b+c)(a+b+c))).
(5)

See also

Mixtilinear Circle, Mixtilinear Incircles

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Cite this as:

Weisstein, Eric W. "Mixtilinear Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MixtilinearTriangle.html

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