A triangle has side lengths a, b, c. Prove that √(a + b - c) + √(b + c - a) + √(c + a - b) ≤ √a + √b + √c. When do you have equality?
Solution
Let A2 = b + c - a, B2 = c + a - b, C2 = a + b - c. Then A2 + B2 = 2c. Also A = B iff a = b. We have (A - B)2 ≥ 0, with equality iff A = B. Hence A2 + B2 ≥ 2AB and so 2(A2 + B2) ≥ (A + B)2 or 4c ≥ (A + B)2 or 2√c ≥ A + B, with equality iff A = B. Adding the two similar relations we get the desired inequality, with equality iff the triangle is equilateral.
© John Scholes
jscholes@kalva.demon.co.uk
11 Apr 2002