1966 IMO Problems/Problem 2
Let , , and be the lengths of the sides of a triangle, and respectively, the angles opposite these sides. If,
Prove that the triangle is isosceles.
We'll prove that the triangle is isosceles with . We'll prove that . Assume by way of contradiction WLOG that . First notice that as then and the identity our equation becomes: Using the identity and inserting this into the above equation we get: Now, since and the definitions of being part of the definition of a triangle, . Now, (as and the angles are positive), , and furthermore, . By all the above, Which contradicts our assumption, thus . By the symmetry of the condition, using the same arguments, . Hence .
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