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distance from a point on diagonal of a cyclic quadrilateral

Source: Sharygin Geometry Olympiad 2017 First Round P22 grades 10-11

July 23, 2018
geometrycyclic quadrilateraldistanceprojection

Problem Statement

Let PP be an arbitrary point on the diagonal ACAC of cyclic quadrilateral ABCDABCD, and PK,PL,PM,PN,POPK, PL, PM, PN, PO be the perpendiculars from PP to AB,BC,CD,DA,BDAB, BC, CD, DA, BD respectively. Prove that the distance from PP to KNKN is equal to the distance from OO to MLML.
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