# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Return Of The Triangle. Topic: Geometry/Inequalities/Trigonometry. Level: AIME. February 11th, 2007

Problem: (1961 IMO – #2) Let be the sides of a triangle, and its area. Prove:

.

In what case does equality hold?

Solution: We begin with the trivial inequality, , which has equality at . Rearrange to get

.

Let be the angle between the sides with lengths . Since (can be proved by combining RHS) with equality at , we know

.

Recalling the Law of Cosines, we know . Also, , so substituting we obtain

as desired. Equality holds when and , which means the triangle must be equilateral. QED.

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Comment: There are lots of ways to prove this, but this is one of the more elementary ones, requiring only basic knowledge of inequalities and trigonometry. Which is always good because I don’t know any geometry. We see that this inequality is in general pretty weak, with equality only when the triangle is equilateral – there is a stronger version that states

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See if you can prove that…

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Practice Problem: Let be the sides of a triangle, and its area. Prove:

.

### 4 Responses to “Return Of The Triangle. Topic: Geometry/Inequalities/Trigonometry. Level: AIME.”

1. blue_giraffe Says:

you do too know geometry

2. t0rajir0u Says:

This is in Engel and they give an IDENTITY (if I recall correctly) from which this follows by trivial inequality. It’s so hardcore.