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Triangle and area

Source: Iran PPCE 2007

March 23, 2007
geometrytrigonometryinequalitiescircumcirclegeometry proposed

Problem Statement

ABCABC is an arbitrary triangle. A′,B′,C′A',B',C' are midpoints of arcs BC,AC,ABBC, AC, AB. Sides of triangle ABCABC, intersect sides of triangle A′B′C′A'B'C' at points P,Q,R,S,T,FP,Q,R,S,T,F. Prove that SPQRSTFSABC=1−ab+ac+bc(a+b+c)2\frac{S_{PQRSTF}}{S_{ABC}}=1-\frac{ab+ac+bc}{(a+b+c)^{2}}
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