An Equilateral Dilemma: Solution
I'd like to thank Hong Li for solving the more interesting part of the problem.
First, lets establish the existence of the outer triangle. We'll demonstrate that at least an equilateral triangle exists.
Let s be the angle depicted above, and s = s' = s'' . Then the lines that pass through the vertices of the inner triangle at the angle s will intersect each other at points P1, P2, P3. Trivially, this forms a triangle, and we are assured they meet since none of the lines generated this way are parallel. By similar triangles, the points form an equilateral triangle.
To demonstrate the triangle formed can only be equilateral, we eliminate the other possibilities.
Isosceles triangles:
Without loss of generality: a = a' != a'' => s = s' != s''
But triangle abc = triangle a'bc, so t = t' => s'' = s', since t + s' = 120, and t' + s'' = 120. A contradiction.
Scalene triangles:
Without loss of generality: a < a' < a'' => s < s' < s''
Similarly, r'' < r < r', when looking at the sidelengths of the large triangle and their opposing angles.
But comparing triangle abc with triangle a'bc we see that s < s' => r > r', since two of the three sides corresponding sides are equal in length. A contradiction.
Thus the outer triangle generated as described must be equilateral.
If you have comments, suggestions or questions, mail me at:
bscriver@speakeasy.org