2001 USAMO Problems/Problem 6
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
We label each upper case point with the corresponding lower case letter as its assigned number. The key step is the following lemma.
Lemma: If is an isosceles trapezoid, then .
Proof: Assume without loss of generality that , and that rays and meet at . Let be the incenter of triangle , and let line bisect . Then is on , so reflecting everything across line shows that is also the incenter of triangle . Therefore, Hence , as desired.
For any two distinct points and in the plane, we construct a regular pentagon . Applying the lemma to isosceles trapezoids and yields Hence . Since and were arbitrary, all points in the plane are assigned the same number.
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