The surprising theorem below by Toshio Seimiya from Japan was communicated to me in July 2018 by my friend and colleague from Poland, Waldemar Pompe:

"In a plane hexagon *ABCDEF*, the lines joining the midpoints of the opposite sides are concurrent, if and only if, triangles *ACE* and *BDF* have equal areas."

Drag any of the red vertices *A*, *B*, *C*, *D*, *E* or *X* to dynamically change the configuration.

This theorem is a generalization of the special case when *ABCDEF* is a hexagon with opposite sides parallel (but not necessarily equal) - see *Parallel-Hexagon Concurrency Theorem*.

*Note*: In certain crossed configurations, the above sketch will unfortunately not correctly show the concurrency of the lines due to a software design decision by the GSP computer programmers regarding the relative movement of points on a line when they are supposed to cross over and interchange. For the general case, one also needs to assume signed areas, which the software also doesn't handle.

**Challenge**:

Although the Seimiya Theorem lends itself most readily to attack by vector methods or complex algebra, the real challenge is to find a purely geometric proof. Can you find such a proof?

**Proof**:

Here's a nice, elegant proof with vectors and coordinates by my long-time friend and colleague, Michael Fox - click *here* to view or download.

**Generalization to 3D**:

Interestingly, and perhaps even more surprisingly, is that though triangles *ACE* and *BDF* having equal areas is a sufficient condition for the Seimiya theorem in the plane, this is not the case in 3D. As shown by the counter-example below, using Cabri 3D, and produced by my friend and colleague from Germany, Heinz Schumann, this is not true for a spatial 3D hexagon. As shown below, triangles *ACE* and *BDF* having equal areas in 3D space, do not necessarily imply that the lines connecting the midpoints of opposite sides are concurrent.

For a spatial '3D-parallelo-hexagon', the forward implication of the Seimiya theorem for a 3D-spatial hexagon follows directly from its point symmetry.

The concept of a 3D-parallelo-hexagon was discussed in a paper by Heinz Schumann & myself in the July 2018 issue of '*The Mathematical Gazette*', and is available to download at: *A surprising 3D result involving a hexagon*. A 3D-parallelo-hexagon is also formed by the centroids of the triangles subdividing any 3D-hexagon as shown at: *Triangle Centroids of a Hexagon form a Parallelo-Hexagon*.

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Michael de Villiers, created 3 January 2019, updated 24 March 2019.