Question 92
Suppose $a, b, c$ are the side lengths of a triangle $ABC$. Let
$$
x = \frac{b+c}{2}, \quad y = \frac{c+a}{2}, \quad z = \frac{a+b}{2}.
$$
Show that $x, y, z$ are the side lengths of a triangle $XYZ$, with the same perimeter as $ABC$, but with a bigger area, unless $ABC$ is equilateral.
$$
x = \frac{b+c}{2}, \quad y = \frac{c+a}{2}, \quad z = \frac{a+b}{2}.
$$
Show that $x, y, z$ are the side lengths of a triangle $XYZ$, with the same perimeter as $ABC$, but with a bigger area, unless $ABC$ is equilateral.