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Journal for Geometry and Graphics 12 (2008), No. 1, 023--034
Copyright Heldermann Verlag 2008

Yff Conics

Clark Kimberling
Dept. of Mathematics, University of Evansville, 1800 Lincoln Avenue, Evansville, IN 47722, U.S.A.


Suppose that $a,b,c$ are algebraic indeterminates and $U=u:v:w$ is a point given in homogeneous trilinear coordinates. The Yff conic of $U$ is defined as the locus of a point $X=x:y:z$ satisfying the equation $f(x,y,z) = f(u,v,w)$, where $f(u,v,w)=(vw+wu+uv)/(u^2+v^2+w^2)$. The symbolic substitution $(a,b,c) \to (bc,ca,ab)$ maps the Yff conic of the symmedian point to that of the centroid. This mapping and others are used to find a large number of special points on many Yff conics.

Keywords: Ellipse, hyperbola, parabola, symbolic substitution, triangle center, trilinear coordinates, trilinear product, Yff conic.

MSC: 51M05

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