The following remark is due to Steiner (Werke, Bd. II, p. 3). Consider the tangents PA, PB to the ellipse ABCD. Consider also the polar AB of P, passing through the tangent points. Define the intersection point Q of the bisector of angle APB with AB. Then, for any point C on the ellipse the line CQ, cuts the ellipse at another point D, such that ang(CQP) = ang(QPB).
The proof follows by realizing that points (C,D,Q,E) form a harmonic division. Then, lines (PC,PD,PQ,PE), passing through these points, form a harmonic bundle of lines. But PQ and PR are, per construction, orthogonal at P. Hence these lines bisect the angle of the two others: ang(CPD).
Steiner, J. Werke Bd. II. New York, Chelsea, 1971, p. 3
Todhunter, I. A treatise on Plane co-ordinate geometry New York, Macmillan and Co. 1888, p. 307.
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