Circumconic and inconic information

In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle,^[1] and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.^[2]

Suppose $A, B, C$ are distinct non-collinear points, and let $△ ABC$ denote the triangle whose vertices are $A, B, C$ . Following common practice, $A$ denotes not only the vertex but also the angle $∠ BAC$ at vertex $A$ , and similarly for $B$ and $C$ as angles in $△ ABC$ . Let $a=|BC|,b=|CA|,c=|AB|,$ the sidelengths of $△ ABC$ .

In trilinear coordinates, the general circumconic is the locus of a variable point $X=x:y:z$ satisfying an equation

uyz+vzx+wxy=0,

for some point $u : v : w$ . The isogonal conjugate of each point $X$ on the circumconic, other than $A, B, C$ , is a point on the line

ux+vy+wz=0.

This line meets the circumcircle of $△ ABC$ in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola.

The general inconic is tangent to the three sidelines of $△ ABC$ and is given by the equation

u^{2}x^{2}+v^{2}y^{2}+w^{2}z^{2}-2vwyz-2wuzx-2uvxy=0.

^ Weisstein, Eric W. "Circumconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Circumconic.html
^ Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Inconic.html

[1] Weisstein, Eric W. "Circumconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Circumconic.html

[2] Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Inconic.html

Circumconic and inconic information

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