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.
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
See if you can prove that…
Practice Problem: Let be the sides of a triangle, and its area. Prove:
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