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tangential and cyclic wanted, area of ABCD equals to MA x MC +MB x MD

Source: Tournament of Towns, Senior A-Level , Fall 2019 p5

April 18, 2020
tangentialcyclic quadrilateralCyclicgeometryareaKvant

Problem Statement

The point MM inside a convex quadrilateral ABCDABCD is equidistant from the lines ABAB and CDCD and is equidistant from the lines BCBC and ADAD. The area of ABCDABCD occurred to be equal to MA⋅MC+MB⋅MDMA\cdot MC +MB \cdot MD. Prove that the quadrilateral ABCDABCD is a) tangential (circumscribed), b) cyclic (inscribed).
(Nairi Sedrakyan)
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