Geodesic
Concurrency and Collinearity in Hexagons
Journal for geometry and graphics, Tome 20 (2016) no. 2, pp. 159-171 Cet article a éte moissonné depuis la source Heldermann Verlag

Voir la notice de l'article

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 
@article{JGG_2016_20_2_JGG_2016_20_2_a1,
     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/}
}
TY  - JOUR
AU  - N. Anghel 
TI  - Concurrency and Collinearity in Hexagons
JO  - Journal for geometry and graphics
PY  - 2016
SP  - 159
EP  - 171
VL  - 20
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/JGG_2016_20_2_JGG_2016_20_2_a1/
ID  - JGG_2016_20_2_JGG_2016_20_2_a1
ER  - 
%0 Journal Article
%A N. Anghel 
%T Concurrency and Collinearity in Hexagons
%J Journal for geometry and graphics
%D 2016
%P 159-171
%V 20
%N 2
%U http://geodesic.mathdoc.fr/item/JGG_2016_20_2_JGG_2016_20_2_a1/
%F JGG_2016_20_2_JGG_2016_20_2_a1
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/