Journal for Geometry and Graphics 20 (2016), No. 2, 159--171
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 non-consecutive sides equals the product of the other three sides. We present here a vast generalization of this result to (closed) hexagonal paths (Sine-Concurrency Theorem), which also admits a collinearity version (Sine-Collinearity Theorem). The two theorems easily produce a proof of Desargues' Theorem. Henceforth we recover all the known facts about Fermat-Torricelli 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, Fermat-Torricelli Point, Napoleon Point, Kiepert Point, Desargues' Theorem, Pascal's Theorem, Brianchon's Theorem.
MSC: 51M04; 51A05, 51N15, 97G70
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