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

This sketch 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.

## 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

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Ceva's Theorem

This above sketch uses **JavaSketchpad**, a World-Wide-Web component of *The Geometer's Sketchpad.* Copyright © 1990-2008 by KCP Technologies, Inc. Licensed only for non-commercial use.

Michael de Villiers, August 2010.