Posts tagged 'stewart'

F.7 Ceva and Menelaus

Three cevians AD, BE, CF of triangle ABC concur at a point P if and only if (AF/FB)(BD/DC)(CE/EA) = +1 (Ceva’s Theorem), whereas points D, E, F’ on the sidelines are collinear if and only if (AF’/F’B)(BD/DC)(CE/EA) = -1 (Menelaus’s Theorem), making (A, B; F, F’) a harmonic range.

F.14 Conway Notation

In John H. Conway’s triangle notation, setting S = 2K (twice the area of triangle ABC), S_omega = (a^2 + b^2 + c^2)/2, and S_A = (b^2 + c^2 - a^2)/2, S_B = (c^2 + a^2 - b^2)/2, S_C = (a^2 + b^2 - c^2)/2 reduces trigonometric and barycentric identities to polynomial expressions satisfying S_B S_C + S_C S_A + S_A S_B = S^2.

F.23 Golden Triangles

In the acute golden isosceles triangle ABC with base AB = 1, legs AC = BC = phi, and angles 72-72-36 degrees, bisecting base angle B meets AC at D such that CD = 1, DA = phi - 1 = 1/phi, and BD = AB = 1, partitioning ABC into a smaller golden triangle BCD and a golden gnomon ABD.

F.46 Stewart’s Theorem

If a cevian of length f from C divides base c = AB into segments k’ c and k c (with k + k’ = 1), Stewart’s Theorem states f^2 = k a^2 + k’ b^2 - k k’ c^2; at the midpoint k = k’ = 1/2, this yields Apollonius’s median formula f^2 = (1/2)(a^2 + b^2 - c^2/2).

F.48 Verisimilitude, Corner Subtriangles, and Excircle Homothety

If lines parallel to the sides of triangle ABC are drawn tangent to the incircle, the three corner subtriangles are homothetic to ABC with scale factors x/s, y/s, z/s (where x = s - a, y = s - b, z = s - c), so their inradii satisfy r_A + r_B + r_C = r.