Routh's theorem
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 formula_1 points formula_2, formula_3, and formula_4 lie on segments formula_5, formula_6, and formula_7, then writing formula_8formula_9, formula_10formula_11, and formula_12formula_13, the signed area of the triangle formed by the cevians formula_14, formula_15, and formula_16 is the area of triangle formula_1 times 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  formula_19 has become popularized as the one-seventh area triangle. The formula_20 case implies that the three medians are concurrent (through the centroid). Proof. Suppose that the area of triangle formula_1 is 1. For triangle formula_22 and line formula_23 using Menelaus's theorem, We could obtain: Then formula_25 So the area of triangle formula_26 is: Similarly, we could know: formula_28 and formula_29 Thus the area of triangle formula_30 is: Citations. The citation commonly given for Routh's theorem is Routh's "Treatise on Analytical Statics with Numerous Examples", Volume 1, Chap. IV, in the second edition of 1896 p. 82 , possibly because that edition has been easier to hand. However, Routh gave the theorem already in the first edition of 1891, Volume 1, Chap. IV, p. 89. Although there is a change in pagination between the editions, the wording of the relevant footnote remained the same. Routh concludes his extended footnote with a "caveat": Presumably, Routh felt those circumstances had not changed in the five years between editions. On the other hand, the title of Routh's book had been used earlier by Isaac Todhunter; both had been coached by William Hopkins. Although Routh published the theorem in his book , that is not the first published statement. It is stated and proved as rider (vii) on page 33 of Solutions of the Cambridge Senate-house Problems and Riders for the Year 1878, i.e., the mathematical tripos of that year, and the link is https://archive.org/details/solutionscambri00glaigoog. It is stated that the author of the problems with roman numerals is Glaisher. Routh was a famous Mathematical Tripos coach when his book came out and was surely familiar with the content of the 1878 tripos examination. Thus, his statement "The author has not met with these expressions for the areas of two triangles that  often occur. " is puzzling. Problems in this spirit have a long history in recreational mathematics and mathematical paedagogy, perhaps one of the oldest instances of being the determination of the proportions of the fourteen regions of the Stomachion board. With Routh's Cambridge in mind, the "one-seventh-area triangle", associated in some accounts with Richard Feynman, shows up, for example, as Question 100, p. 80, in "Euclid's Elements of Geometry (Fifth School Edition)", by Robert Potts (1805--1885,) of Trinity College, published in 1859; compare also his Questions 98, 99, on the same page. Potts stood twenty-sixth  Wrangler in 1832 and then, like Hopkins and Routh, coached at Cambridge. Pott's expository writings in geometry were recognized by a medal at the International Exhibition of 1862, as well as by an Hon. LL.D. from the College of William and Mary, Williamsburg, Virginia.