# 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: 1. Draw the long diagonal AE, and draw circles at either end, with radius AE. 1. Mark H, the point of intersection of the two circles, and draw intervals HA and HE. 1. Draw a line parallel to HE through C, and mark I, the point of intersection with GA. 1. Draw a line parallel to HA through C, and mark J, the point of intersection with EF. Draw interval IJ. 1. Shade $\triangle$CEI, the required equilateral triangle. I submit the following example of the construction for an irregular polygon ... and make the observations:

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

2. 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.

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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• I'm not sure about your very first sentence. Couldn't only $2$ or even $1$ be on the edge? – Jack M Jul 31 '14 at 0:12
• Remember that the polygon is convex, so if the triangle's corner points are within the polygon then all points along the edges must also be within the polygon. – NovaDenizen Jul 31 '14 at 0:13
• If only 1 vertex is on the edge then you can certainly make the triangle bigger by pushing the other two out to an edge. – NovaDenizen Jul 31 '14 at 0:16
• It's also an interesting problem to find the largest equilateral triangle with three touching vertices. – Dávid Horváth Apr 3 '18 at 5:40

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. :-/ (Image from this link.)

• It is not in Math Reviews either. Can you post a pdf? – i. m. soloveichik Jul 31 '14 at 0:47
• @i.m.soloveichik: I can eventually, but not soon, as it requires a visit to physical archives & scanning. – Joseph O'Rourke Jul 31 '14 at 1:12
• I've asked about a special case (probably trivial!) of this question here: Finding the largest equilateral triangle inside a given triangle – ShreevatsaR Aug 3 '17 at 16:02
• I got ahold of that paper and uploaded it here. – Michael L. Aug 4 '17 at 4:02
• On the second page of the paper, proof of Theorem 1 says “Let $e_a$, $e_b$ and $e_c$ be the three edges of $P$ that contain the corners $a$, $b$ and $c$ of $T$ respectively”. But from the second diagram on the linked question, it appears that not all 3 corners of the inscribed equilateral triangle $T$ need be on edges of $P$. Am I missing something? – ShreevatsaR Aug 4 '17 at 4:30