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The return of American geo

Source: USAJMO 2023/6

March 23, 2023
AMCUSA(J)MOUSAJMOgeometry

Problem Statement

Isosceles triangle ABCABC, with AB=ACAB=AC, is inscribed in circle ω\omega. Let DD be an arbitrary point inside BCBC such that BD≠DCBD\neq DC. Ray ADAD intersects ω\omega again at EE (other than AA). Point FF (other than EE) is chosen on ω\omega such that ∠DFE=90∘\angle DFE = 90^\circ. Line FEFE intersects rays ABAB and ACAC at points XX and YY, respectively. Prove that ∠XDE=∠EDY\angle XDE = \angle EDY.
Proposed by Anton Trygub
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