**NOTE**: Please WAIT while the applets below load.

**Conjecture**: Trisect the sides of a triangle *ABC* as shown in the figure below. Then *DG*, *EH* and *FI* are concurrent in *X*. Drag any of *A*, *B* or *C*.

A side trisection triangle concurrency

**Challenge**: Can you *explain why* (prove that) the conjecture is true?

**Explore more**: In the interactive sketch *here*, with *P*, *Q*, *R*, *S*, *T*, and *U* the midpoints of the shown segments, some additional properties of the same geometric configuration are shown (which include a circumscribed ellipse as well as an inscribed ellipse of the hexagon *DEFGHI*).

**Additional challenge**: Can you *explain* (prove) these further conjectures? What happens if sides are divided into four or more equal parts? Explore!

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Created by Michael de Villiers, 6 November 2013, modified 14 November 2015.