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All lines touch same circle

Source: Tournament of Towns oral round p2

March 21, 2016
geometryLocuscircles

Problem Statement

On plane there is fixed ray ss with vertex AA and a point PP not on the line which contains ss. We choose a random point KK which lies on ray. Let NN be a point on a ray outside AKAK such that NK=1NK=1. Let MM be a point such that NM=1,MāˆˆPKNM=1,M \in PK and M!=K.M!=K. Prove that all lines NMNM, provided by some point KK, touch some fixed circle.
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