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Prove that the radius of the circle..

Source: Indian IMO Training camp-2011

May 17, 2011
geometrytrapezoidtrigonometrycyclic quadrilateralgeometry unsolved

Problem Statement

Let ABCABC be a triangle each of whose angles is greater than 30∘30^{\circ}. Suppose a circle centered with PP cuts segments BCBC in T,Q;CAT,Q; CA in K,LK,L and ABAB in M,NM,N such that they are on a circle in counterclockwise direction in that order.Suppose further PQK,PLM,PNTPQK,PLM,PNT are equilateral. Prove that:
a)a) The radius of the circle is 2abca2+b2+c2+43S\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S} where SS is area.
b)a⋅AP=b⋅BP=c⋅PC.b) a\cdot AP=b\cdot BP=c\cdot PC.
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