# Erdos-Mordell Inequality

Problem 1

Let $$M$$ be the point inside the triangle $$ABC$$. Denote by $$A_1$$, $$B_1$$, $$C_1$$ the feet of perpendiculars from $$M$$ to $$BC$$, $$CA$$, and $$AB$$. Prove that $MA\cdot MB\cdot MC\geq 8 MA_1\cdot MB_1\cdot MC_1.$

Problem 2 (Nairi Sedrakyan, IMO 1996)

Let $$ABCDEF$$ be a convex hexagon such that $$AB$$ is parallel to $$DE$$, $$BC$$ is parallel to $$EF$$, and $$CD$$ is parallel to $$AF$$. Let $$R_A,R_C,R_E$$ be the circumradii of triangles $$FAB,BCD,DEF$$ respectively, and let $$P$$ denote the perimeter of the hexagon. Prove that $R_A+R_C+R_E\geq\frac{P}2.$

Theorem 1 (Erdos-Mordell)

Let $$M$$ be the point inside the triangle $$ABC$$. Denote by $$A_1$$, $$B_1$$, $$C_1$$ the feet of perpendiculars from $$M$$ to $$BC$$, $$CA$$, and $$AB$$. Prove that $MA+MB+MC\geq 2(MA_1+MB_1+MC_1).$

Problem 3

If $$M$$ is a point inside the triangle $$ABC$$ and if $$A_1$$, $$B_1$$, $$C_1$$ are feet of perpendiculars from $$M$$ to $$BC$$, $$CA$$, $$AB$$, prove that $\frac1{MA}+\frac1{MB}+\frac1{MC}\leq \frac12\left(\frac1{MA_1}+\frac1{MB_1}+\frac1{MC_1}\right).$