IrMO 2010 Problem 10

Suppose $a,b,c$ are the side lengths of a triangle $ABC$ . Show that $$x=\sqrt{a(b+c-a)}, y=\sqrt{b(c+a-b)}, z=\sqrt{c(a+b-c)}$$ 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.