The following interesting result was experimentally discovered in 2018 by my long time mentor and colleague Piet Human (retired from the University of Stellenbosch, now at the Ukuqonda Institute) while he was developing some elementary geometry materials about lines for primary school learners.
Initial Conjecture
Draw 3 rays from a point J to intersect 2 rays drawn from a point O and label the intersections as shown below. Then the intersection I of the diagonals of quadrilateral ECFB, the intersection H of the diagonals of quadrilateral ADFC are collinear with the point O.
Explore
Drag any of the red vertices/points to change the configuration below to explore and verify the conjecture above. Also click on the provided "Show Additional Properties" button. What do you observe?
Challenge
1) Can you explain why (prove that) the above conjectures are true?
2) Click on this Show Hint button if necessary.
Published Paper
Read the joint 2019 paper by myself & Piet Human in Learning & Teaching Mathematcs, No. 26, pp. 28-30, published by AMESA, available at: An Interesting Collinearity.
Some Related Links
Triangle Incentre-Circumcentre Collinearity
Some Trapezoid (Trapezium) Explorations (see Investigations 3 & 4)
Four Collinear Points
Quadrilateral Similar Triangles Collinearity (An application of Desargues & Pappus)
Concurrency, collinearity and other properties of a particular hexagon
Euler line proof (using transformations)
Desargues' Theorem
Some Bicentric Quadrilateral Properties
Collinear Conjecture
Investigating incentres of some iterated triangles
Investigating circumcentres of iterated median triangles
Nine Point Conic and Generalization of Euler Line
Further Euler line generalization
Spieker Conic and generalization of Nagel line
The quasi-circumcentre and quasi-incentre of a quadrilateral
The quasi-Euler line of a quadrilateral and a hexagon
A side trisection triangle concurrency
Side Divider (Wind Screen Wiper) Theorem for a Tangential Quadrilateral
Angle Divider Theorem for a Cyclic Quadrilateral
Conway’s Circle Theorem as special case of Side Divider Theorem
A 1999 British Mathematics Olympiad Problem and its dual (Concurrency)
External Links
Pappus's hexagon theorem (Wikipedia)
Desargues's theorem (Wikipedia)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, created with WebSketchpad, 23 February 2019; updated 26 March 2026.