Half-Altitude Circle

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Half-AltitudeCircle

The half-altitude circle is the circumcircle of the half-altitude triangle. The circle function is given by

(1)

where

f(a,b,c)=5a^8-8a^6b^2-6a^4b^4+16a^2b^6-7b^8-8a^6c^2+28a^4b^2c^2-16a^2b^4c^2-4b^6c^2-6a^4c^4-16a^2b^2c^4+22b^4c^4+16a^2c^6-4b^2c^6-7c^8,

(2)

which is not a Kimberling center.

The center has center function

(3)

where

g(a,b,c)=-2a^(10)-a^8b^2+12a^6b^4-10a^4b^6-2a^2b^8+3b^(10)-a^8c^2-16a^6b^2c^2+10a^4b^4c^2+16a^2b^6c^2-9b^8c^2+12a^6c^4+10a^4b^2c^4-28a^2b^4c^4+6b^6c^4-10a^4c^6+16a^2b^2c^6+6b^4c^6-2a^2c^8-9b^2c^8+3c^(10),

(4)

which is not a Kimberling center. The radius is given by

R=sqrt((h(a,b,c)h(b,c,a)h(c,a,b))/((-a+b+c)(a-b+c)(a+b-c)(a+b+c)))1/(32a^2b^2c^2|cosAcosBcosC|),

(5)

where

h(a,b,c)=2a^6-2a^4b^2-2a^2b^4+2b^6-3a^4c^2+6a^2b^2c^2-3b^4c^2+c^6.

(6)

No Kimberling centers lie on the half-altitude circle.

See also

Central Circle, Half-Altitude Triangle

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Cite this as:

Weisstein, Eric W. "Half-Altitude Circle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Half-AltitudeCircle.html

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