Geometric inequalities (Table of contents)

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 \[\frac1{MA}+\frac1{MB}+\frac1{MC}\leq \frac12\left(\frac1{MA_1}+\frac1{MB_1}+\frac1{MC_1}\right).\]