Convex Equal-Area Polygons (CEAP)

Definition

A plane polygon is called Equal-Area when all triangles formed by 3 consecutive vertices have the same area (see [1]). This property is invariant by Affine Transformations – when applying an Affine Transformation to a polygon, all areas are multiplied by a common factor, so they remain equal to each other.

Highlighted areas are equal to each other

We are interested in Convex Equal-Area polygons with N sides. What do they look like?

• N=3: every triangle is CEA. Unremarkable. L
• N=4: a CEA quadrilateral must be a parallelogram.
• N=5: every CEA pentagon is affinely equivalent to the regular one!
• N≥5: discounting affine transformations, there are N-5 degrees of freedom in your choice of a CEAP ([1] or [2].

Construct your own CEA decagon

You can use the Applet below (created with Geogebra) to construct your own 10-sided CEA polygon!

This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com

You can move P₁, P₂ and P₃ at will (but they just apply an affine transformation to the whole figure).
You can move P₄, P₅, P₆, P₇ and P₈ along the dotted lines (5 degrees of freedom).
From P₄ on, each vertex only affects the ones with higher indices.
The applet will determine P₉ and P₁₀ in order to close a convex equal-area polygon.

How does it work?

There are two keys to the construction of P₉ and P₁₀:

• The hyperbola h through P₇, with asymptotes r₁=P₁P₈ and r₂=P₆P₉, must have a second branch going through P₁₀.
• If P₃, I and K are lined up in this order, the correct branch of h intersects r₃ (usually at two points) if and only if area(IJK) ≥4 area(P₁P₂P₃).
• If I, P₃ and K are lined up in this order, the correct branch of h intersects r₃ at one point, which is P₁₀ (the alternative intersection does not give us a convex polygon).

Open Questions

·         Given a strictly convex curve C, a point P₁ on it, and a number N, we can approximate C by a CEAP passing through P₁ (essentially doing the construction above; details in [2]). Now, can we always find an N-sided CEAP passing through P₁ inscribed in C?
[We believe the answer is “yes” if N is odd – but we think the answer is “no” for even N!]

• Starting from a regular polygon, and moving the vertices continuously, can we get to any CEAP – keeping it CEA throughout the whole process? In other words, is the space of all N-sided CEAPs connected?
  [We believe the answer is “yes”.]

References

[1] G.Harel and J.M.Rabin, “Polygons whose vertex triangles have equal area”; Amer. Math. Monthly 110 (2003), 606-619.

[2] M. Craizer, R. Teixeira and M. Horta, “Affine properties of convex equal-area polygons”; Discrete & Computational Geometry, October 2012, Volume 48, Issue 3, pp 580-595.