I was trying to prove for myself the assertion that the locus

of the incenter over all triangles with given Euler segment is

the interior of the orthocentroidal circle, or "arena". However,

the algebra turns out to be a bit too tough even for me - I can

prove that the incenter is always inside this circle, but not that

all points of it are reached.

So I hope that one of the hyacinths (Paul perhaps?) can help by

finding the expression for the squared distance from the incenter

to the midpoint of GH as a function of a,b,c.

I'm also interested in the locus of the excenters. I thought

at first that they would all lie outside the arena, but seem to

have disproved this. There are two problems:

1) what's the set of points where any excenter can be?

2) supposing a > b > c, what are the individual loci of

the a-, b-, and c- excenters?

I suspect that the answer to 1) has a fairly simple shape,

while those that arise in 2) might be quite complicated.

Regards to all, John Conway