5

Part of 2003 Austrian-Polish Competition

Problems(1)

(x+y+z)^2 <= (a^2+b^2+c^2)/2 + 2S.sqrt(3), in a triangle?

Source: a Geo book!

A triangle with sides a, b, c has area S. The distances of its centroid from the vertices are x, y, z. Show that: if (x + y + z)^2 ≤ (a^2 + b^2 + c^2)/2 + 2S√3, then the triangle is equilateral.

geometryinequalitiesgeometry solved