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


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