Geodesic
Concurrency and Collinearity in Hexagons
Journal for geometry and graphics, Tome 20 (2016) no. 2, pp. 159-171

Voir la notice de l'article provenant de la source Heldermann Verlag

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.
Classification : 51M04, 51A05, 51N15, 97G70
Mots-clés : Hexagon, concurrency, collinearity, Fermat-Torricelli Point, Napoleon Point, Kiepert Point, Desargues' Theorem, Pascal's Theorem, Brianchon's Theorem 
N. Anghel. Concurrency and Collinearity in Hexagons. Journal for geometry and graphics, Tome 20 (2016) no. 2, pp. 159-171. http://geodesic.mathdoc.fr/item/JGG_2016_20_2_JGG_2016_20_2_a1/
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     author = {N. Anghel},
     title = {Concurrency and {Collinearity} in {Hexagons}},
     journal = {Journal for geometry and graphics},
     pages = {159--171},
     year = {2016},
     volume = {20},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JGG_2016_20_2_JGG_2016_20_2_a1/}
}
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