[alogo] A theorem of Steiner

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Consider the secants XY of a circle b, which are parallel to the fixed direction a. The circles with diameters XY are tangent to an ellipse c. The ellipse has small axis x = AB the diameter parallel to a. Its great axis is y = sqrt(2)*x. Actually only the circles with XY >= y are tangent. The circles with XY <y are entirely inside the ellipse. Notice that the ellipse c is characterized by the fact that its vertices A, B and the foci C, D are on the same circle b.

Steiner Werke, bd I, p.176 (Vorgelegte Aufgaben und Lehrsaetze, 2 Lehrsatz).
Give a geometric (non-analytic) proof.

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