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Part of 2024 Oral Moscow Geometry Olympiad

Problems(2)

Isosceles trapezoid with perpendicular diagonals

Source: Oral Moscow Geometry Olympiad 2024, 8-9.1

9/3/2024
In an isosceles trapezoid, the diagonals are perpendicular. Find the distance from the center of the circle described around the trapezoid to the point of intersection of its diagonals, if the lengths of the bases are equal to aa and bb.
geometrytrapezoid
chord of circle construction tangent to an ellpse

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

11/29/2023
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
conicsellipseconstructiongeometrybmoeg