Problem:
As shown in the figure on the right, is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of .
Solution:
The key to the solution is the fact that if two triangles have the same height, then their areas are proportional to their bases. Therefore, from the figure below, we have the equations
Solving the first two of these equations simultaneously, one finds that and . After checking that these values also satisfy the third equation, one may conclude that the area of is or .
The problems on this page are the property of the MAA's American Mathematics Competitions