1. Given an equilateral triangle and a point in its interior which lies at distances of 3 units, 4 units, and 5 units from the vertices of the triangle, find the length of a side of the triangle.
This is an old chestnut which has an ingenious solution. It is included primarily to motivate:
2. In the problem above, the solution in not rational.
a. First, given an equilateral triangle of side d and a point a units, b units, and c units from the vertices of the triangle, write down an equation relating a, b, c, and d.
b. Use the techniques of this talk to try to find integer solutions to this equation. [Hint #1: You might first consider solutions to configurations of this sort
Hint #2: What are the singular points which arise in this problem?]