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x^2 + xy + y^2 = a^2 , y^2 + yz + z^2 = b^2, z^2 + zx + x^2 = c^2

Source: Polish MO Recond Round 1985 p1

September 9, 2024
algebrainequalitiesGeometric Inequalitiesgeometry

Problem Statement

Inside the triangle ABCABC, the point PP is chosen. Let a,b,c a, b, c be the lengths of the sides BC BC , CA CA , AB AB , respectively, and x,y,z x, y, z the distances of the point P P from the vertices B,C,A B, C, A . Prove that if x2+xy+y2=a2 x^2 + xy + y^2 = a^2 y2+yz+z2=b2y^2 + yz + z^2 = b^2 z2+zx+x2=c2z^2 + zx + x^2 = c^2 this a2+ab+b2>c2. a^2 + ab + b^2 > c^2.
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