16 Jun
2007
16 Jun
'07
2:29 p.m.
On Saturday 16 June 2007 14:31, Joshua Zucker wrote:
On 6/16/07, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
Given triangle ABC, A' divides side BC in the ratio of m:1, B' divides CA in the ratio n:1, and C' divides AB in the ratio p:1. Now draw cevians AA', BB', and CC'. The ratio of the area of the center triangle to the area of the original triangle is: f(m,n,p) = (mnp-1)^2/((mn+n+1)(mp+m+1)(np+p+1))
Lovely. Is there an equally lovely proof?
I was wondering what symmetry this expression needs to have. I think I see why f(m,n,p) = f(1/n, 1/m, 1/p). Are there other symmetries I'm missing?
Cyclic permutation of m,n,p. -- g