Do you know about Erdos-Mordell Inequality , it is a geometric inequality .
Let be a point inside , draw the feets of perpendicular of to the sides , namely .
Then we always have :
The inequality holds when it is an equilateral triangle , being the centre , that means given a nonequilateral triangle , we can never find a point satisfying the equality .
By applying this , we have
where denotes the inradius and the incentre .
Therefore , , the equality holds when
It is a very useful inequality. Look here.
For the purpose of cracking Olympiad problems, knowing Arithmetic Mean - Geometric Mean inequality, Cauchy Schwarz inequality and Jensen's inequalty helps a lot.
Look these up in wikipedia. Various applications of Schur's inequality and Muirhead's inequality are also frequently seen in Olympiad problems.