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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.
Can you make a conjecture about the point X? Which special point in the triangle is it?
A side trisection triangle concurrency
Challenge
1) Can you explain why (prove that) the concurrency conjecture is true? Can you explain why (prove that) point X is the centroid?
2) Can you prove it in different ways? Can you use Varignon's theorem to prove the result?
3) Can you generalize further?
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).
Published Paper
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).
Related Links
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva)
Point Mass Centroid (centre of gravity or balancing point) of Quadrilateral
Centroid of Cardboard (Lamina) Quadrilateral
Generalizations involving maltitudes
Triangle Centroids of a Hexagon form a Parallelo-Hexagon
More Properties of a Bisect-diagonal Quadrilateral
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
An associated result of the Van Aubel configuration and some generalizations
Generalizations of a theorem of Sylvester (about forces acting on a point in a triangle)
Napoleon's Theorem: Generalizations & Converses
The 120o Rhombus (or Conjoined Twin Equilateral Triangles) Theorem
Jha and Savaran’s generalisation of Napoleon’s theorem
Dao Than Oai’s generalization of Napoleon’s theorem
Euler line proof
Nine Point Conic and Generalization of Euler Line
A further generalization of the Euler line
Spieker Conic and generalization of Nagel line
Generalizing the Nagel line to Circumscribed Polygons by Analogy
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
The quasi-Euler line of a quadrilateral and a hexagon
External Links
Centroid (Wikipedia)
Napoleon's theorem
Free Download of Geometer's Sketchpad
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Created by Michael de Villiers, 6 November 2013, modified to WebSketchpad, 14 November 2015; 20 March 2024.