CollinearityDate: 04/20/99 at 12:42:57 From: Jonathan Hirsch Subject: Geometry Do you know of a point that is collinear with the incenter of a triangle (the intersection of the angle bisectors), and the Gergonne point (the intersection of AIa, BIb, and CIc)? Thanks. Date: 04/20/99 at 14:35:31 From: Doctor Floor Subject: Re: geometry Hi, Jonathan, Thanks for your question! The most notable points in a triangle collinear with the incenter and the Gergonne point are the De Longchamps point, the isoperimetric point, and the equal detour point. The De Longchamps point is the reflection of the orthocenter (the intersection of the altitudes) through the circumcenter (the intersection of the perpendicular bisectors). The isoperimetric point is the point P in a triangle ABC, such that triangles ABP, BPC, ACP have equal perimeter. It was shown to exist in triangles fulfilling certain conditions by G. R. Veldkamp in 1985 (in the American Mathematical Monthly). The equal detour point is the point X that equalizes the detour when you travel from a vertex to another vertex via X (for example, the detour from A to B equals |AX|+|XB|-|AB|). Under certain conditions the equal detour point is not the only point having this condition, since then the isoperimetric point has the equal detour property too. This second point is described in the same article by G. R. Veldkamp. Surprisingly, these "sophisticated points" lie on this simple line. For more on these two modern triangle centers, see this page by Professor Clark Kimberling of the University of Evansville: http://cedar.evansville.edu/~ck6/tcenters/recent/isoper.html Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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