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right triangle from AC,BD,CD if <ADB + <ACB = <CAB + <DBA = 30^o, AD = BC.

Source: 2021 Grand Duchy of Lithuania, Mathematical Contest p3 (Baltic Way TST) https://artofproblemsolving.com/community/c1321893_grand

October 30, 2021
geometryright triangleangles

Problem Statement

Let ABCDABCD be a convex quadrilateral satisfying ∠ADB+∠ACB=∠CAB+∠DBA=30o\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^o, AD=BCAD = BC. Prove that there exists a right-angled triangle with side lengths ACAC, BDBD, CDCD.
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