Talk:Triangle center/Archive 1

Archive 1

Excenter

The excenters ARE triangle centers. It's simply a matter of defining the right functions. If

 f(a,b,c) = 1 if a < b or a < c ; otherwise -1
 g(a,b,c) = 1 if b < c or b < a ; otherwise -1
 h(a,b,c) = 1 if c < a or c < b ; otherwise -1

then f(a,b,c):g(a,b,c):h(a,b,c) is the excenter opposite the largest angle of the triangle. MRFS (talk) 10:11, 20 June 2009 (UTC)

The function f(a,b,c) as defined above is not homogeneous.

Let a < b and t > 0. Then ta < tb and so f(a,b,c)=1 and f(ta, tb, tc) = 1.

Let a < b and t < 0. Then ta > tb and so f(a,b,c) = 1 and f(ta, tb, tc) = − 1.

Thus there is no n such that f(ta, tb, tc) = tⁿf(a,b,c) and hence the function f(a,b,c) is not homogeneous. Therefore the point f(a,b,c) : g(a,b,c) : h(a,b,c) is not a triangle center. Krishnachandranvn (talk) 01:55, 21 June 2009 (UTC)

OK. I was making the (sensible) assumption that t > 0. The domain of f is the subset of R³ where a,b,c are the sides of a triangle; in other words they are non-negative and satisfy the triangle inequality. If t < 0 then (ta,tb,tc) is outside this domain so f(ta,tb,tc) isn't properly defined. The formal definition should explicitly state what range of values t can have.

However even if negative values of t are permitted a trivial change to the example makes f homogeneous.

f(a,b,c) = 1 if |a| < |b| or |a| < |c| ; otherwise -1

g(a,b,c) = 1 if |b| < |c| or |b| < |a| ; otherwise -1

h(a,b,c) = 1 if |c| < |a| or |c| < |b| ; otherwise -1

The constant n is of course zero so that tⁿ is always unity.

MRFS (talk) 11:59, 21 June 2009 (UTC)

Whether or not the excenters can be homogenously defined for an equilateral triangle, they surely exist. Merely because a particular system of coordinates cannot describe the points does not mean that they cease to be well-defined in other systems. hgilbert (talk) 10:26, 27 September 2010 (UTC)

Of course the excenters can be homogeneously defined for an equilateral triangle! In trilinears they are simply -1:1:1, 1:-1:1, and 1:1:-1. However they cannot be represented by a triangle center function because a=b=c implies f(a,b,c)=f(b,c,a)=f(c,a,b). Since this always reduces to 1:1:1 it means there is no possible way to represent the excenters (or indeed any point other than 1:1:1) using a triangle centre function. The recent edit needs corrected or undone. MRFS (talk) 12:59, 30 September 2010 (UTC)

Meaning of the definition?

In geometry a, triangle center is a point in the plane whose trilinear coordinates relative to a reference triangle are functions of the side-lengths of that triangle having the properties of homogeneity, bisymmetry and cyclicity.

Picturing this doesn't come instantly to me. I'd expect any reasonable definition to imply equivariance with respect to isometries of the plane. Does this imply that? Could be, but I think that if so, it should say so. Michael Hardy (talk) 23:07, 8 July 2009 (UTC)

I think that defining a point in terms of trilinear coordinates ensures that the definition is equivariant with respect to direct isometries; bisymmetry implies equivarience w.r.t. reflections; homogeneity implies invariance w.r.t. homogenous dilations; and cyclicity implies equivariance w.r.t. cyclic permutations of vertices. So not only are triangle centres equivariant w.r.t. isometries, they are also equivariant w.r.t. the wider group of similarities. There is some discussion of this near the end of the article, but I agree it could be made more prominent. Gandalf61 (talk) 14:10, 30 September 2010 (UTC)

At the time Michael's comments were written the above definition was (as he said) difficult to picture. The article did not even mention equivariance or similarity. It was drastically remodelled on 18 July 2009 to meet his criticisms, and justification for the revised definition was included. Perhaps it should be more prominent but it was shunted down to the end because it's slightly technical. It's impossible to please everyone! MRFS (talk) 10:17, 1 October 2010 (UTC)

Ah, yes, have just realised I was replying to a comment posted over a year ago. Note to self - always check date tags. Gandalf61 (talk) 12:14, 1 October 2010 (UTC)