The dynamic geometry activities below are from my book Rethinking Proof (free to download).
Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Worksheet & Teacher Notes: Concurrency.
Concurrency
Given two lines in the plane, they can either be parallel or meet in a point. But given three lines in the plane, they can either be:
1) all three parallel to each other
2) two parallel to one another, and the third one oblique
3) none are parallel to each other, in which case, they either form:
a) a triangle, or
b) are concurrent (meet in one point).
Concurrent lines, and even other concurrent geometric objects like circles, parabola, etc. are highly interesting and have been an important productive research area for geometers since ancient times.
In the activity below you will be led to make a conjecture about the concurrency of three lines and investigate whether the conjecture is true or not.
Concurrency Conjecture
Notes
1) This is another conjecturing activity from the Verification section of Rethinking Proof.
2) What do you notice about the lines AE, BF and CD in the sketch above?
3) Are they always concurrent? Check by dragging.
4) Click on the 'Link to Checking' button to navigate to a new sketch where you can check your answer to 2-3 above.
5) Note that this activity is intended as a prelude to the next Rethinking Proof activity which is about the Concurrency of the Altitudes.
References
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. & Heideman, N. (2014). Conjecturing, refuting and proving within the context of dynamic geometry. Learning & Teaching Mathematics, no. 17, pp. 20-26.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Related Links
Triangle Altitudes (Rethinking Proof activity)
Water Supply II: Three Towns (Rethinking Proof activity)
The Center of Gravity of a Triangle (Rethinking Proof activity)
The Fermat-Torricelli Point (Rethinking Proof activity)
Airport Problem (Rethinking Proof activity)
Napoleon (Rethinking Proof activity)
Miquel (Rethinking Proof activity)
Area Ratios (Rethinking Proof activity)
Investigating incentres of some iterated triangles
Collinear Conjecture
Kosnita's Theorem
Fermat-Torricelli Point Generalization (aka Jacobi's theorem) plus Further Generalizations
Dual to Kosnita (De Villiers points of a triangle)
Another concurrency related to the Fermat point of a triangle plus related results
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva's theorem)
Point Mass Centroid (centre of gravity or balancing point) of Quadrilateral
Bride's Chair Concurrency
Triangle Centroids of a Hexagon form a Parallelo-Hexagon: A generalization of Varignon's Theorem
Nine-point centre & Maltitudes of Cyclic Quadrilateral
A side trisection triangle concurrency
A 1999 British Mathematics Olympiad Problem and its dual
Concurrency and Euler line locus result
Haag Hexagon - Extra Properties
Concurrency, collinearity and other properties of a particular hexagon
Power Lines of a Triangle
Carnot's Perpendicularity Theorem & Some Generalizations
Generalizing the concepts of perpendicular bisectors, angle bisectors, medians and altitudes of a triangle to 3D
Anghel's Hexagon Concurrency theorem
Some Circle Concurrency Theorems
Three Overlapping Circles (Haruki's Theorem)
Parallel-Hexagon Concurrency Theorem
Toshio Seimiya Theorem: A Hexagon Concurrency result
Van Aubel's Theorem and some Generalizations (See concurrency in Similar Rectangles on sides)
The quasi-circumcentre and quasi-incentre of a quadrilateral
External Links
Refutation in a Dynamic Geometry Context (Sine of the Times)
Concurrent lines (Wikipedia)
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, 26 July 2025; updated 27 July 2025.