18

Part of 2008 Sharygin Geometry Olympiad

Problems(1)

Inequality with S

Source: Sharygin contest 2008. The correspondence round. Problem 18

(A.Abdullayev, 9--11) Prove that the triangle having sides

a

b

c

S

satisfies the inequality a^2\plus{}b^2\plus{}c^2\minus{}\frac12(|a\minus{}b|\plus{}|b\minus{}c|\plus{}|c\minus{}a|)^2\geq 4\sqrt3 S.

inequalitiesgeometrytrigonometrygeometry proposed