TITLE: Largest Equilateral Triangle in a Polygon
QUESTION [6 upvotes]: Is there an algorithm to determine the largest equilateral triangle in a convex polygon?

REPLY [2 votes]: 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.