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cosA+cosB+cosC+cosD =0 => ABCD is cyclic or trapezoid

Source: Czech and Slovak Olympiad 1984, National Round, Problem 2

September 11, 2024
geometrytrigonometrytrapezoid

Problem Statement

Let α,β,γ,δ\alpha, \beta, \gamma, \delta be the interior angles of a convex quadrilateral, If cos⁡α+cos⁡β+cos⁡γ,+cos⁡δ=0, \cos\alpha + \cos\beta + \cos\gamma, + \cos\delta = 0 , then this quadrilateral is cyclic or a trapezium. Prove it.
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