The Center of Gravity of a Triangle

The dynamic geometry activities below are from the "Proof as Explanation" section of 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: The Center of Gravity of a Triangle.

Recommendation
a) It is highly recommended that students, teachers or parents use cardboard triangles, and other cardboard polygons, in conjuction with the dynamic geometry investigations below. In order to locate the approximate center of gravity (balancing point) of a cardboard polygon, students can be asked to balance the polygon on the tip of a pencil or other sharp-pointed object. There is educational value in then having them compare their experimental findings with the theoretical locations of the centers of gravity (balancing points).
b) Also in conjuction with the below, view video clips of the dynamic simulations by the gravity simulator of Cinderella (free to download) of other ways of finding the centroids (centers of gravity or balance) of a triangle (with weights at the vertices) at:
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices

The Center of Gravity of a Triangle

Notes
1) Users are strongly encouraged to download the accompanying Worksheet, and to print it out to use in conjunction with the dynamic sketch above. Alternatively, copy the URL: http://dynamicmathematicslearning.com/centre-of-gravity-rethinking-proof.pdf and paste it into a new browser window. Then resize the new window to place it side by side with this one, or one below the other. (However, this is not likely to be a feasible option for users using small screens such as a cellphone or tablet.)
2) In the first activity in the sketch above, the path (locus) traced out by the midpoint of DE as D is dragged along the side AB is called a median.
3) Use the tools on the left to accurately construct all three medians of the triangle. What do you notice? Check your observation by dragging any of the vertices of the triangle.
4) Click on the 'Link to Centroid Properties' button to navigate to a new sketch which shows some of the important properties of the centroid of a triangle.
5) Click on the 'Measurement' buttons to display some measurements. What do you notice? Check your observations by dragging.
6) Challenge: Can you explain why (prove that) your observations in 3) and 5) are always true?
7) To prove that the medians are concurrent and that the centroid G divides each of the medians in the ratio 1:2, click on the 'Construction for Proof' button to display an auxiliary construction that will help you with your logical explanation/proof.
8) Next complete the guided explanation (proof) in the accompanying Worksheet - also the one explaining (proving) why the areas of the six triangles are equal.
9) Question 1, Further Exploration of Worksheet: to locate the centroid of a cardboard quadrilateral, go to: Centroid of Cardboard (Lamina) Quadrilateral.
10) Next look at the following webpages involving two other centroids for quadrilaterals, namely, Point Mass (Vertex) Centroid of Quadrilateral and Centroid (balancing point) of Perimeter Quadrilateral.
11) Question 3, Further Exploration of Worksheet: Use the tools on the left (scroll down if necessary) to measure the x and y coordinates of the vertices of your triangle as well as its centroid. Calculate the average of the x and y coordinates of the vertices. What do you notice? Can you generalize further?

Alternative Median Concurrency Proof using Areas
12) Click on the 'Link to Alternative Proof' button to navigate to a new sketch and click the 'Step' buttons to view a proof of the concurrency of the medians using areas ratios.
13) Note that in the last step (Step 7), by carefully looking back at the given proof, reveals that product of the area ratios is always equal to 1, even if the points D, E and F are not midpoints.
14) This area proof therefore naturally leads to Ceva's theorem, and provides a useful pedagogical example of the discovery function of proof (De Villiers, 1998, pp. 381-382). You can navigate to it by clicking on the 'Link to Ceva' button.
15) More-over, the converse of Ceva's theorem is also true and provides a powerful tool for proving the concurrency of lines in a triangle - click on 'Link to Ceva converse' button to navigate to a sketch that provides an experimental confirmation of its truth. Now use the converse of Ceva's theorem to prove the concurrency of the following cevians in a triangle:
.....a) medians
.....b) angle bisectors
.....c) altitudes.
.....d) lines from the vertices of a triangle to the corresponding points on opposite sides where the incircle touches.
16) Apart from Ceva's theorem, the concurrency of the medians of a triangle can be seen as a special case of the Similar Isosceles △'s concurrency theorem (see the The Fermat-Torricelli Point). It can also be quite easily proved by other advanced geometry theorems. For example, see if you can use the following advanced theorems yourself:
.....a) Homothetic Polygons Concurrency Theorem
.....b) Desargues' Theorem of two perspective triangles.
17) Ceva's theorem is also used and applied in several of the dynamic links given further down, most notably in the following concurrency proofs:
.....Fermat-Torricelli Point Generalization (aka Jacobi's theorem)
.....Three Overlapping Circles (Haruki's Theorem)
.....Six Point Cevian Circle.
18) The trigonometric form of Ceva's Theorem is often very useful in certain problem solving contexts.
19) Anghel (2018) has generalized the trigonometric version of Ceva's theorem to hexagons - for a dynamic version, see Anghel's Hexagon Concurrency theorem
(clicking on the 'Show Anghel's Hexagon Concurrency theorem' button in the opening sketch).

Using Scientific Principles
20) Kogan (1974, p. 6) provides a novel proof using forces to prove the concurrency of the medians.
21) The method of balancing weights in conjuction with the law of the lever is a powerful, explanatory way to explain & prove not only the concurrency of the medians, but also Ceva's theorem - see Balancing Weights in Geometry as a Method of Discovery & Explanation
22) The same balancing weights method is also useful in deriving Varignon's theorem (see Hanna & Jahnke, 2002), with the centre of the Varignon parallelogram giving us point mass (vertex) centroid of a quadrilateral.
Historical Note
A proof of the concurrency of the medians does not appear in Euclid's Elements (300 BC), and the first known proof of the concurrency of the medians of a triangle is generally attributed to Archimedes (c. 287–212 BC) in his work On the Equilibrium of Planes. Archimedes' approach was rooted in physics and the concept of the center of mass (centroid). He proved that the center of mass of a uniform triangular lamina must lie on each of its medians - just like we showed in the opening slide above. Since the center of mass is a single, unique point, this demonstrates that all three medians must pass through that same point, thus proving their concurrency - also compare with Balancing Weights in Geometry.

References
Anghel, N. (2018). Concurrency in Hexagons - a Trigonometric Tale. Journal for Geometry and Graphics, Volume 22, No. 1, 21–29.
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. (1994, 1996, 2009). Some Adventures in Euclidean Geometry (free to download). Dynamic Mathematics Learning.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Hanna, G. & Jahnke, H.N. (2002). Arguments from Physics in Mathematical Proofs: An Educational Perspective. For the Learning of Mathematics, Vol. 22, No. 3 (Nov.), pp. 38-45.
Kogan, B. Yu. (1974). The Application of Mechanics to Geometry. The University of Chicago Press, Chicago, IL 60637.
Oliver, P.N. (2001). Pierre Varignon and the Parallelogram Theorem. Mathematics Teacher, Vol. 94, No. 4, April, pp. 316-319.

Other Rethinking Proof Activities
Other Rethinking Proof Activities

External Links
Median (geometry) (Wikipedia)
Ceva's theorem (Wikipedia)
Trigonometric Form of Ceva's Theorem (Cut The Knot)
Homothety (Wikipedia)
Desargues's theorem (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
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, 25 Nov 2025; 8 Jan 2026; 14 Feb 2026.