In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle points , , and lie on segments , , and , then writing , , and , the signed area of the triangle formed by the cevians , , and is
where is the area of the triangle .
This theorem was given by Edward John Routh on page 82 of his Treatise on Analytical Statics with Numerous Examples in 1896. The particular case has become popularized as the one-seventh area triangle. The case implies that the three medians are concurrent (through the centroid).
Proof
Routh's theorem
Suppose that the area of triangle is 1. For triangle and line using Menelaus's theorem, We could obtain:
Then
So the area of triangle is:
Similarly, we could know: and
Thus the area of triangle is:
