A side trisection triangle concurrency

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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.