Marion's Theorem

Marion's theorem (Mathematics Teacher 1993, Maushard 1994, Morgan 1994) 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.

This 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.

A generalization of Marion's theorem sometimes known as Morgan's theorem was found by Ryan Morgan, a sophomore at Patapsco High School in Baltimore (Morgan 1994). If the sides of the triangle are instead partitioned into equal segments for an odd integer and each division point in connected to the opposite vertex, a central hexagon is still formed (Maushard 1994). Morgan's theorem states that this hexagon has area

(25)

relative to the original triangle (Morgan 1994). For , 3, 5, ..., this gives one over the centered nonagonal numbers 1, 10, 28, 55, 91, 136, 190, 253, 325, 406, ... (OEIS A060544).