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chord of circle construction tangent to an ellpse

Source: 2023 BMoEG III p6 https://artofproblemsolving.com/community/c594864h3181962p28994736

November 29, 2023
conicsellipseconstructiongeometrybmoeg

Problem Statement

In a plane: 1. An ellipse with foci F1F_1, F2F_2 lies inside a circle ω\omega. Construct a chord ABAB of ω\omega. touching the ellipse and such that AA, BB, F1F_1, and F2F_2 are concyclic. 2. Let a point PP lie inside an acute angled triangle ABCABC, and A′A', B′B', C′C' be the projections of PP to BCBC, CACA, ABAB respectively. Prove that the diameter of circle A′B′C′A'B'C' equals CPCP if and only if the circle ABPABP passes through the circumcenter of ABCABC.
Proposed by Alexey Zaslavsky
https://cdn.artofproblemsolving.com/attachments/8/e/ac4a006967fb7013efbabf03e55a194cbaa18b.png
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