The lively siblings of the Pentagon theorem
Tóm tắt
The five circles in the classical Pentagon theorem of Miquel are given as circumcircles of five certain triangles in the pentagon. If one chooses instead the circumcircles of five other triangles, one gets a different configuration of circles. This resulting configuration of circles carries three families of five concyclic quadruples of points. Together with the five circumcircles this gives a total of 20 circles. The radical axes of each two of these twenty circles are all concurrent.
Tài liệu tham khảo
Halbeisen, Lorenz, Hungerbühler, Norbert, Loureiro, Vanessa: The Pentagon theorem in Miquelian Möbius planes. Submitted for publication (2023)
Halbeisen, Lorenz, Hungerbühler, Norbert, Loureiro., Vanessa: The hidden twin of Morley’s five circles theorem. Amer. Math. Monthly, to appear
Li, Hongbo, Xu, Ronghua, Zhang, Ning: On Miquel’s five-circle theorem. In: Li, Hongbo, Olver, Peter J., Sommer, Gerald (eds.) Computer algebra and geometric algebra with applications, pp. 217–228. Springer, Berlin Heidelberg (2005)
Miquel, Auguste: Théorèmes de géométrie. J. Math. Pures Appl. 3, 485–487 (1838)