Collinearity

Date: 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/