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Modlova 3rd tst, problem 1

Source: Moldova TST III

March 26, 2006
geometrycircumcircletrigonometryinequalitiesincentergeometry proposed

Problem Statement

Let the point PP in the interior of the triangle ABCABC. (AP,(BP,(CP(AP, (BP, (CP intersect the circumcircle of ABCABC at A1,B1,C1A_{1}, B_{1}, C_{1}. Prove that the maximal value of the sum of the areas A1BCA_{1}BC, B1ACB_{1}AC, C1ABC_{1}AB is p(R−r)p(R-r), where p,r,Rp, r, R are the usual notations for the triangle ABCABC.
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