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Let a, b, c be the sides of a triangle with semi-perimeter S. Given that b*c/(b+c) + c*a/(c+a) + a*b/(a+b) = S, what type of triangle must it be?
- Equilateral
- Scalene
- Isosceles
- Right isosceles
Correct answer: Equilateral
Solution
Semi-perimeter S = (a+b+c)/2. Each term bc/(b+c) <= (b+c)/4 by AM-HM. Summing: LHS <= (a+b+c)/2 = S, with equality iff a = b = c. So the given condition is satisfied if and only if the triangle is equilateral.
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