<s>[INST] Sides of triangle and an altitude Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Let $h$ be the altitude drawn on the side of length $a$   Then is  $a^2 + 4h^2 - (b+c)^2$ always negative ?
 [/INST] Let's suppose the triangle is acute  

We have
$$
a_1^2 + h^2 = b^2\\
a_2^2 + h^2 = c^2
$$
and by Cauchy-Schwarz inequality
$$
a_1\cdot a_2 + h\cdot h \leq \sqrt{a_1^2 + h^2}\sqrt{a_2^2 + h^2} = bc
$$
Summing the above relations we get
$$
a^2 + 4h^2 = a_1^2 + a_2^2 + 2a_1 a_2 + 4h^2 \leq b^2 + c^2 +2bc = (b + c)^2
$$
A similar reasoning shows the inequality is true even for obtuse triangles.
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