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).

Read our Dec. 2013 paper "A trisection concurrency: a variation on a median theme" (with Shunmugam Pillay) in the Learning and Teaching Mathematics journal in pdf format (881 kb) or in iBook format (2 MB).


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