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Excentral-Hexyl Ellipse

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ExcentralHexylEllipse

The excentral-hexyl ellipse is the ellipse passing through vertices of the excentral and hexyl triangles (P. Moses, pers. comm., Jan. 29, 2005). It has center at the circumcenter O. It has trilinear equation

a(a+b-c)(a-b+c)(b+c)alpha^2+2abc(a+b+c)betaalpha+2abc(a+b+c)gammaalpha-b(a-b-c)(a+b-c)(a+c)beta^2 -(a+b)(a-b-c)c(a-b+c)gamma^2+2abc(a+b+c)betagamma.
(1)

The ellipse has area

A=4piabc[((a+b)(b+c)(c+a))/(a+b+c)]^(3/2)Delta ×product_(cyclic)1/(sqrt(f(a,b,c))),
(2)

where

f(a,b,c)=a^4-ba^3-b^2a^2-2c^2a^2+bca^2+b^3a+bc^2a+2b^2ca+c^4-bc^3-b^2c^2+b^3c.
(3)

The only Kimberling center the ellipse passes through is X_(1768).

See also

Bevan Circle, Circumellipse, Excentral Triangle, Hexyl Triangle

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Cite this as:

Weisstein, Eric W. "Excentral-Hexyl Ellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Excentral-HexylEllipse.html

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