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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.