Routh's theorem

Statement

Let $ABC$ be a triangle in the Euclidean plane with area $S$ . Let $D$ , $E$ , $F$ be points in the interiors of the sides $BC$ , $CA$ , $AB$ , and let $x = BD/DC$ , $y = CE/EA$ and $z = AF/FB$ . Then the three segments $AD$ , $BE$ , $CF$ bound a triangle whose area is $S \cdot \frac{(xyz - 1)^2}{(xy + x + 1)(yz + y + 1)(zx + z + 1)}.$ When $xyz = 1$ , this triangle degenerates to a point.

Remarks

A proof places the triangle in barycentric coordinates, computes the three intersection points of the cevians, and takes the ratio of determinants. Example: with $x = y = z = 2$ , the area is $S \cdot 49/343 = S/7$ , the well-known one-seventh triangle. The case $xyz = 1$ is exactly the condition of Ceva's theorem for the three cevians to be concurrent, and Routh's formula gives area $0$ there.

Going deeper

For a student

Routh's theorem refines Ceva's theorem: Ceva says when the inner triangle is a point, and Routh measures it in all other cases. Because affine maps preserve ratios of areas and ratios along lines, it is enough to prove it for one triangle, for example $A = (0, 0)$ , $B = (1, 0)$ , $C = (0, 1)$ . The theorem is a good exercise in barycentric coordinates and in the use of the determinant to compute areas.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Triangle concept
Area of a triangle concept
Routh's theorem statement

Tag	`VWQQY7NFED`
URI	`m2://VWQQY7NFED`
Kind	statement, theorem
Writing	W0 stub
MSC	51M04, 51M25
Status	This copy of the site has no build data. Run `m2 status VWQQY7NFED`.
About	Triangle
Uses	Triangle, Area of a triangle
Same as	Wikidata Q1789863 Wikipedia Routh's theorem