Question 87
Suppose $a, b, c$ are the side lengths of a triangle $ABC$. Show that
$$
x = \sqrt{a(b+c-a)}, \quad y = \sqrt{b(c+a-b)}, \quad z = \sqrt{c(a+b-c)}
$$
are the side lengths of an acute-angled triangle $XYZ$, with the same area as $ABC$, but with a smaller perimeter, unless $ABC$ is equilateral.
$$
x = \sqrt{a(b+c-a)}, \quad y = \sqrt{b(c+a-b)}, \quad z = \sqrt{c(a+b-c)}
$$
are the side lengths of an acute-angled triangle $XYZ$, with the same area as $ABC$, but with a smaller perimeter, unless $ABC$ is equilateral.