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min AB: CD wanted, diagonals AC, BD bisect DX, CX respectively in cyclic ABCD

Source: 2019 Croatia MO p7

August 2, 2020
ratiogeometryCyclicbisects segmentcyclic quadrilateral

Problem Statement

On the side ABAB of the cyclic quadrilateral ABCDABCD there is a point XX such that diagonal ACAC bisects the segment DXDX, and the diagonal BDBD bisects the segment CXCX. What is the smallest possible ratio ∣AB∣:∣CD∣|AB | : |CD| in such a quadrilateral ?
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