They're different except for the equilateral triangle, I think, @heather. It turns out that there's a cool result about orthocenter, centroid, and incenter. They're all collinear (and the one in the middle divides the segment in a 2:1 ratio)!!!
Oh, cool. @Alessandro. Just wanted to know the context. OK. It's easy if you've done a particular G&P exercise (which is covered in the my text in a theorem).
Yes, @heather ... given any 2 you can find the third.
I've done most of the exercises on page 11-13 so I'll start doing those now. I was thinking about arguing that this set is not a submanifold of $\Bbb R^2$ using the implicit function theorem but let's see if I find something useful in this exercises
so, the reason i asked was this: finding the orthocenter is a bit of a pain, so i wanted to know if finding the incenter was easier, so that you could find the incenter and centroid (which is really easy to find, by finding the average of the vectors) and then use the relation between the three to find the orthocenter.
but, i actually don't know how to find the equation of a line that bisects an angle.
Actually, writing down perpendicular bisectors is easier. But there are easier ways to find the orthocenter (e.g., using a little bit of linear algebra). This stuff is more conceptual and not necessarily meant for calculating. That said, you can do both incenter and orthocenter quite easily with straightedge and compass.
@Alessandro: I meant that equation to describe the $y$-axis. It is a manifold. But the implicit function theorem fails.
$f(x,y)=x^2$
Sorry you're sick, @Brody. I escaped ATL 3 days early for fear of icemageddon, thereby enraging my friends in Athens whom I never went to see. As you well know, there was nary a flake and only a little ice.
@Astyx let $f$ be the the distance function for Alice from the point Y with {0,x} as the domain. let $g$ be the function that is the distance for Bob from Y. f(1)=0, $0\leq g(1)<x$. By IVT $f(d)=g(d)$ for some $d\in${0,x}.
@TedShifrin Oh, that sucks... or maybe not so much for you? I don't know how it was down in ATL but in upper Cobb it snowed just a bit Friday night and the following morning. That's all, but the constant refreezing ice still spun commuters off roads this morning.
Well, I was remembering a few winters ago when I was stuck in my house in AThens for over 5 days. I didn't want a repeat of that. I'm never visiting ATL again other than in spring or fall. :) Get better soon!
how are the sides of a "circle" called that is up and down, if you would tinker a circle out of paper? and how is the side called that points towards the middlepoint of a circle and how is the side called that points in the opposite direction?
