Marion's Theorem

Marion's theorem, also known as Marion Walter's theorem (Education Development Center 2003), states that the area of the central hexagonal region determined by trisection of each side of a triangle and connecting the corresponding points with the opposite vertex is given by 1/10 the area of the original triangle. The theorem is named after mathematics educator Marion Walter and was first published by Cuoco et al. (1993).

The area ratio can easily be shown using trilinear coordinates. In the above diagram, , , and, from the multisection formula, the trisection points have trilinear coordinates

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)

The other labeled points can then be computed as

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)

Using the trilinear equation for the area of a triangle then gives the following areas of the colored triangles illustrated above in terms of the area of the original triangle.

			(19)
			(20)
			(21)
			(22)

Taking the remaining red portion then gives

			(23)
			(24)

as originally stated.

Morgan's theorem gives an odd-subdivision generalization of Marion's theorem.

Kazakov (2026) considered a higher-dimensional analogue in which every edge of an -simplex is trisected and, for each edge, the two hyperplanes determined by one of the two trisection points and the vertices not on the edge are taken. In barycentric coordinates, the central polytope is the set of points satisfying

(25)

and has relative -dimensional volume

(26)

This gives for , for , and for .