An Equilateral Dilemma: Solution
The original problem
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.
|Let's start with the Cosine Law. With the above variables we have:|
|Rearrange to isolate cos r:|
|Identify cos r - cos r':|
|Express a' relative to a:|
|Cancel common terms:|
|Applying the same cosine law above between t' and t" we get (after skipping identical steps):|
Therefore the outer triangle generated as described must be equilateral.
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