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Triangles with Equal Area

Source: 2010 IrMO Paper 2 Problem 5

January 30, 2018
geometry

Problem Statement

Suppose a,b,ca,b,c are the side lengths of a triangle ABCABC. Show that x=a(b+c−a),y=b(c+a−b),z=c(a+b−c)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 XYZXYZ, with the same area as ABCABC, but with a smaller perimeter, unless ABCABC is equilateral.
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