This page uses **JavaSketchpad**, a World-Wide-Web component of *The Geometer's Sketchpad.* Copyright © 1990-2001 by KCP Technologies, Inc. Licensed only for non-commercial use.

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IsogonalConjugate

In the plane of a triangle *ABC*, the isogonal conjugate of a line is a conic section. Specifically, it is an ellipse, parabola, or hyperbola according as the line meets the circumcircle in 0, 1, or 2 points. Here, isogonal conjugates of points on line *DE* are points on an ellipse. For example, the isogonal conjugate of point *F* is point *G*.
The point *X* between *F* and *G* determines the ratio |*FX*|/|*FG*|. This ratio is used to dilate points on line *DE* to a set of intermediate points, which are linked by segments to give the appearance of an intermediate curve. When *X* is animated, the curve represents "slow motion isogonal conjugation" of the line into the conic.

This is a sensitive sketch, but with care, you can drag *A*, *B*, *C*, *F*, *D*, and *E* so as to examine a variety of ellipses and hyperbolas.

For more, see *GEOMETRY IN ACTION*, Chapter 7, Project 6.