I maintain working into Edward John Routh (1831–1907). He’s finest recognized for the Routh-Hurwitz stability criterion however he pops up sometimes elsewhere. The earlier put up mentioned Routh’s mnemonic for moments of inertia and his “stretch” theorem. This put up will talk about his triangle theorem.
Earlier than stating Routh’s theorem, we have to say what a cevian is. Giovanni Ceva (1647–1734) was an Italian geometer, finest recognized for Ceva’s theorem, and for a development in that theorem now referred to as a cevian.
A cevian is a line from the vertex of a triangle to the alternative facet. Draw three cevians by connecting every vertex of a triangle to a degree on its reverse facet. If the cevians intersect at some extent, Ceva’s theorem says one thing about how the strains divide the edges. If the cevians type a triangle, Routh’s theorem discover the realm of that triangle.
Routh’s theorem is a generalization of Ceva’s theorem as a result of if the cevians intersect at a typical level, the realm of the triangle fashioned is zero, after which Routh’s space equation implies Ceva’s theorem.
Let A, B, and C be the vertices of a triangle and let D, E, and F be the factors the place their cevians intersect the alternative sides.
Let x, y, and z be the ratios into which all sides is split by the cevians. Particularly let x = FB/AB, y = CD/BD, and z = EA/CE.
Then Routh’s theorem says the realm of the inexperienced triangle fashioned by the cevians is
If the cevians intersect at some extent, the realm of the triangle is 0, which suggests xyz = 1, which is Ceva’s theorem.