## Monday, January 2, 2012

### 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 DE, and F lie on segments BCCA, and AB, then writing $\tfrac{CD}{BD}$ x$\tfrac{AE}{CE}$ y, and $\tfrac{BF}{AF}$ z, the signed area of the triangle formed by the cevians ADBE, and CF is the area of triangle ABC times
$\frac{(xyz - 1)^2}{(xy + y + 1)(yz + z + 1)(zx + x + 1)}.$
This theorem was given by Edward John Routh on page 82 of his Treatise on Analytical Statics with Numerous Examples in 1896.

Source: Wikipedia