
Journal for Geometry and Graphics 20 (2016), No. 2, 159171 Copyright Heldermann Verlag 2016 Concurrency and Collinearity in Hexagons Nicolae Anghel Dept. of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX 76203, U.S.A. anghel@unt.edu In a cyclic hexagon the main diagonals are concurrent if and only if the product of three mutually nonconsecutive sides equals the product of the other three sides. We present here a vast generalization of this result to (closed) hexagonal paths (SineConcurrency Theorem), which also admits a collinearity version (SineCollinearity Theorem). The two theorems easily produce a proof of Desargues' Theorem. Henceforth we recover all the known facts about FermatTorricelli points, Napoleon points, or Kiepert points, obtained in connection with erecting three new triangles on the sides of a given triangle and then joining appropriate vertices. We also infer trigonometric proofs for two classical hexagon results of Pascal and Brianchon. Keywords: Hexagon, concurrency, collinearity, FermatTorricelli Point, Napoleon Point, Kiepert Point, Desargues' Theorem, Pascal's Theorem, Brianchon's Theorem. MSC: 51M04; 51A05, 51N15, 97G70 [ Fulltextpdf (1236 KB)] 