## Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices

Please enable Java for an interactive construction (with Cinderella).

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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## Ceva's Theorem

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: CP, AQ and BR are concurrent, if and only if, the product of the splits are: [B/A][C/B][A/C] = 1 or equivalently (AP/PB)*(BQ/QC)*(CR/RA) = 1

#### .sketch_canvas { border: medium solid lightgray; display: inline-block; } Ceva's Theorem

Challenge
1) Can you prove Ceva's theorem?
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).

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Created by Michael de Villiers, August 2010; updated 16 October 2021.