IMO 1967 LL POL39

Show that the triangle whose angles satisfy the equality

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IMO 1967 LL POL39

Origin: POL

Problem

Show that the triangle whose angles satisfy the equality sin2 A + sin2 B + sin2 C cos2 A + cos2 B + cos2 C = 2 is a right-angled triangle.

Solution

Since sin2 A + sin2 B + sin2 C + cos2 A + cos2 B + cos2 C = 3, the given equality is equivalent to cos2 A+cos2 B+cos2 C = 1, which by multiplying by 2 is transformed into 0 = cos2A + cos 2B + 2 cos2 C = 2 cos(A + B) cos(A −B) + 2 cos2 C = 2 cosC(cos(A −B) −cos C). It follows that either cos C = 0 or cos(A −B) = cos C. In both cases the triangle is right-angled.