Problem

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Tags: geometry, geometric inequality



Let $ABC$ be an acute-angled triangle and variable $D \in [BC]$ . Let's denote by $E, F$ the feet of the perpendiculars from $D$ to $AB$, $AC$ respectively . a) Show that $$\frac{4S^2}{b^2+c^2}\le DE^2 + DF^2\le max \{h_B^2 + h_C^2 \}.$$b) Proved that, if $D_0 \in [BC]$ is the point where the minimum of the sum $DE^2 + DF^2$ is achieved, then $D_0$ is the leg of the symmetrical median of $A$ facing the bisector of angle $A$. c) Specify the position, of $D \in [BC]$ for which the maximum of the sum $DE^2 + DF^2$ is achieved. (The area of the triangle $ABC$ was denoted by $S$ and $h_b, h_c$ are the lengths of the altitudes from $B$ and $C$ respectively)