Routh's Theorem

Move ABC to determine the shape of a desired triangle and move the slider. What do you observe?

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The Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the intersection of three cevians. The theorem states that if in triangle ABC points D, E, and F lie on segments BC, CA, and AB, then writing  = x,  = y, and  = z, the signed area of the triangle formed by the cevians AD, BE, and CF is the area of triangle ABC times

This theorem was given by Edward John Routh on page 82 of his Treatise on Analytical Statics with Numerous Examples in 1896.


Source: Wikipedia


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Rectangle-Triangle Area Relationship

Consider the rectangle and the triangle below.

1.  What do you observe? What are common properties between the two polygons?
2. Drag point D to change the height of the triangle and drag point A or B to change the length of the base. What do you observe?
3. What can you say about the areas of the two figures? Justify your answer. 
4. Click the Show Areas check box in the applet.  Was your answer in 3 correct? 
5. Drag point E. What is the relationship between the area of the rectangle and the area of a triangle? Explain why this is so. 

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

The applet above demonstrates that the area of a triangle is half the area of a rectangle with the same base and height.

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The Dancing Triangle

Move the vertices to determine the shape of your triangle and observe what happens.

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1. What do you observe when you move the three vertices?
2. Is there a relationship between the movement of the three vertices and the area of the triangle?
3. Move point R. What do you observe?
4. Explain why your observation in 3 is such. 

Download GGB file here.