For each point in the plane a real number is assigned such that for every triangle its incenter is assigned the mean of the numbers assigned to its vertices.

Prove that the same number is assigned to every point in the plane.

Begin with a degenerate triangle where ABC are collinear. The incenter is B (or whichever is between the others).

http://www.mathopenref.com/triangleincenter.html
Drag B onto the segment AC to see this.

So we have (A+B+C)/3 = B
(A+C)/3 = 2B/3
(A+C) = 2B
B=(A+C)/2

Now B can be any point between A and C.  So all the points on the segment are equal.   Choosing A and C arbitrarily far apart means every number on the line is equal.

If this is true for ever line in the plane it is clearly true for the whole plane.

Posted by Jer on 2011-03-23 13:31:32