Largest Equilateral Triangle in a Polygon Is there an algorithm to determine the largest equilateral triangle in a convex polygon?

Based on my other answers to more specific tasks, I offer the following solution that will work for any regular convex polygon:



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*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:


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*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.
