Suppose ABC is an equilateral triangle and P is a point inside the triangle, such that PA = 3 cms., PB = 4 cms., and PC = 5 cms.
Then find the length of the side of the equilateral triangle.

I made one construction:

ABC and rotate it 60 degrees clockwise.
Then PAP' is an equilateral triangle of side length 4, and PP'C is a 3-4-5 triangle, meaning AP'C has degree measure 150, and P'C=3, and P'A=4.

By law of cosines, the side length = sqrt(25 + 12sqrt(3))

Posted by tom on 2003-12-20 00:00:36