Largest Equilateral Triangle in a Polygon Is there an algorithm to determine the largest equilateral triangle in a convex polygon?
 A: Based on my other answers to more specific tasks, I offer the following solution that will work for any regular convex polygon:



*

*Draw the long diagonal AE, and draw circles at either end, with radius AE.





*Mark H, the point of intersection of the two circles, and draw intervals HA and HE.





*Draw a line parallel to HE through C, and mark I, the point of intersection with GA.





*Draw a line parallel to HA through C, and mark J, the point of intersection with EF. Draw interval IJ.





*Shade $\triangle$CEI, the required equilateral triangle.



I submit the following example of the construction for an irregular polygon ...

and make the observations:


*

*the inscribed equilateral triangle of maximum area within an irregular polygon will have one side on the polygon's longest side.

*the inscribed equilateral triangle of maximum area within a regular polygon will have as it's axis of symmetry the longest axis of symmetry of the polygon.
A: My initial assumption is wrong, and an 80-80-20 triangle provides the coutnerexample.

If the triangle is maximal then all 3 vertices have to be on edges of the polygon.  So I'm thinking that it makes sense to approach the problem by iterating over all tuples of 3 edges (filtering out tuples where all 3 edges are the same), assuming one vertex is somewhere on each of these edges, and seeking solutions using each tuple of edges.
Say $x_1$, $x_2$, and $x_3$, are the vertices of an equilateral triangle. 
If $x_i$ is constrained to be on a line segment between $p_i$ and $q_i$, then $x_i = p_i \alpha_i + (1 - \alpha_i)q_i$ where $0 \le \alpha_i \le 1$ is an interpolation variable that puts each $x_i$ at a proportion of $\alpha_i$ along each line segment.
Define $v_1$, $v_2$, and $v_3$ as the vector edges of this triangle ($v_1 = x_1 - x_2$, $v_2 = x_2 - x_3$, $v_3 = x_3 - x_1$).  We have $v_1 \cdot v_1 = v_2 \cdot v_2 = v_3 \cdot v_3$ since the lengths of the sides must be equal.  Substitute the definitions of $v_i$ into these equations, then substitute the interpolation equations for $x_i$ above into that.
Now the equations look like $P_1(\alpha_1, \alpha_2, \alpha_3) = P_2(\alpha_1, \alpha_2, \alpha_3) = P_3(\alpha_1, \alpha_2, \alpha_3)$ where each $P_i$ is a 2nd degree multinomial.  All solutions of these multinomials obeying the $\alpha$ range requirement form equilateral triangles touching the edge of the polygon.
Any particular instance of these equations could have 0, 1, or 2 solutions, maybe some other number, and possibly an infinite range of solutions (like a regular (or sufficiently symmetrical) polygon with 3n edges).  If you find a finite number of solutions you can just test them to see which results in the largest edge size.  If you find an infinite range of solutions then finding a maximal triangle is more complicated.
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A: Yes, there is an algorithm:

DePano, A., Yan Ke, and J. O’Rourke. "Finding largest inscribed equilateral triangles and squares." Proc. 25th Allerton Conf. Commun. Control Comput. 1987.

Unfortunately, I do not have easy access to my own article. :-/

 
 
 

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