The Cinderella sketch above simulates how one can experimentally find the centroid ('balancing point') of a triangle with different weights at the vertices.
Instructions: Press the PLAY button to start.Then press in succession, the STEP 1, STEP 2 and STEP 3 buttons. After the triangle and plumb line has come to rest you can also drag the vertices of the triangle, the plumb line or change the weight at the vertices. Press the STOP button to reset.
Created with Cinderella by Uli Kortenkamp & slightly modified by Michael de Villiers, July 2010.
Video Clip
Since the Cinderella sketch above is based on Java, it is unfortunately not likely to run any more on newer computers & new browsers. However, below is a video clip illustrating the sketch.
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The above simulation is a physical demonstration of a theorem named after an Italian mathematician named Giovanni Ceva (1648-1734) who published his theorem in 1678 and proving it by considering centers of gravity and the law of moments. In his honour the line segments joining the vertices of a triangle to any given points on the opposite sides, are called cevians.
Ceva's Theorem
The lines CP, AQ and BR are concurrent, if and only if, the product of the 'splits' are: or equivalently:
Ceva's Theorem
Challenge
1) Can you prove Ceva's theorem?
Solution: A proof of Ceva's theorem is given on pp. 381-382 of De Villiers (1998) by generalizing an explanatory proof of the concurrency of the mediands of a triangle. Ap proof is also given in the Teacher Notes of the 'The Center of Gravity of a Triangle' activity from my Rethinking Proof book (De Villiers, 1999).
2) Can you apply Ceva's theorem to easily prove the concurrency of the medians, angle bisectors and altitudes of a triangle?
3) Use Ceva's theorem to prove that the lines, from the vertices of a triangle to the corresponding points on opposite sides where the incircle touches, are concurrent (in the Gergonne point).
References
De Villiers, M. (1998). An Alternative Approach to Proof. From Lehrer, R. Chazan, D. Designing Learning Environments for Developing Understanding of Geometry and Space. Mahwah, NJ: Lawrence Erlbaum Publishers, pp. 369-393.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Related Links
Experimentally Finding the Medians and Centroid of a Triangle"
The Center of Gravity of a Triangle (Rethinking Proof activity)
Water Supply II: Three Towns (Rethinking Proof activity)
Triangle Altitudes (Rethinking Proof activity)
The Fermat-Torricelli Point (Rethinking Proof activity)
Airport Problem (Rethinking Proof activity)
Napoleon (Rethinking Proof activity)
Miquel (Rethinking Proof activity)
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
Point Mass Centroid (centre of gravity or balancing point) of Quadrilateral
Power Lines of a Triangle
Power Lines Special Case: Altitudes of a Triangle
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
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
Ceva's theorem (Wikipedia)
Trigonometric Form of Ceva's Theorem (Cut The Knot)
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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Go to the related "Experimentally Finding the Medians and Centroid of a Triangle"
Back to "Dynamic Geometry Sketches"
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Created by Michael de Villiers, August 2010; updated 16 October 2021; 6 October 2025.