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 
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     author = {N. Anghel },
     title = {Concurrency and {Collinearity} in {Hexagons}},
     journal = {Journal for geometry and graphics},
     pages = {159--171},
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     number = {2},
     year = {2016},
     url = {http://geodesic.mathdoc.fr/item/JGG_2016_20_2_JGG_2016_20_2_a1/}
}
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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/