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Show Tangency of Circles

Source: Ukrainian Mathematical Olympiad 2023. Day 1, Problem 11.3

April 5, 2023
geometrytangency

Problem Statement

In the quadrilateral ABCDABCD ∠ABC=∠CDA=90∘\angle ABC = \angle CDA = 90^\circ. Let P=AC∩BDP = AC \cap BD, Q=AB∩CDQ = AB\cap CD, R=AD∩BCR = AD \cap BC. Let ℓ\ell be the midline of the triangle PQRPQR, parallel to QRQR. Show that the circumcircle of the triangle formed by lines AB,AD,ℓAB, AD, \ell is tangent to the circumcircle of the triangle formed by lines CD,CB,ℓCD, CB, \ell.
Proposed by Fedir Yudin
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