1982 USAMO Problems/Problem 3
If a point is in the interior of an equilateral triangle and point is in the interior of , prove that
where the isoperimetric quotient of a figure is defined by
First, an arbitrary triangle has isoperimetric quotient (using the notation for area and ):
Lemma. is increasing on , where .
Proof. is increasing on the desired interval, because is increasing on
Let and be half of the angles of triangles and in that order, respectively. Then it is immediate that , , and . Hence, by Lemma it follows that Multiplying this inequality by gives that , as desired.
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