point A. Point D describes a curve called hypopede of Proclus. Its equation was found in Hippopede.html and is:

(x

The following properties are well known or can be easily proved [Mathcurve, Hippopede].

1) The curve is the envelope of circles whose centers H are on the ellipse with focus at A and its symmetric A w.r. to O. The circles are

also constrained to pass through the center of the ellipse. The major axis of this ellipse is R and its minor axis is b = |AS| = sR,

determined by the position of point A inside the circle O(R).

2) The medial line BG of OD is tangent to the previous ellipse at a point H of it.

3) The tangent at D to the hippopede is the reflexion on BG of the tangent at O of the generating circle passing through D.

4) The inversion with respect to circle O(R) maps the ellipse onto the curve resulting from the hippopede by a π/2-rotation about O.

5) The Pedal curve of the ellipse with respect to its center O is the hippopede described by point G. This curve is homothetic to the

original hippopede by the ratio 1/2.

See the derivation of the equation in Hippopede.html [Mathcurve, Hippopede], [Seggern, p. 88], [Booth, II, p. 163].

[Mathcurve] Robert Ferreol

[Seggern] David von Seggern

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