Would there be any problems with riding a fixed-gear road bike in a tri, as long as it had two working brakes?

Flames not appreciated, I don't want advice on buying a $3000 TT or tri bike.

It's not illegal as far as I know... so go for it...

I don't think so. Recumbents are out but anything with a standard geometry is OK.

Uhhhh... trying to run afterwards?

A friend of mine tried to talk me into riding fixed-gear for Ironman Wisconsin. This bike course has a lot of short steep hills. I thought about it for about 5 seconds and realized I could probably get through the 112 mile ride but that my legs would be so toasted from grunting up the hills and spinning like a madman down the hills that I would suffer even more than normal for the run. I could probably be talked into riding fixed for a flat triathlon, though.

I was going to do it in a team triathlon - I'm not a very skilled swimmer and not really built for running (whenever I tried it in the past, my knees did not agree). I have a friend who is comfortable with half-marathons and the occasional marathon, we just have to find a swimmer by next summer. I have yet to find out which triathlons allow teams to enter, but I don't plan on doing more than 1/2 Ironman at most (56mi). From the research I've done so far, "Olympic Distance" seems in my range, but by next summer who knows!
Anyway, thanks to you all for the input, it is much appreciated. I really only ride fixed gear these days, and didn't want to have to buy beg steal or borrow a road bike! I'm planning on reinstalling a rear brake and putting on aero bars, and that will probably be it.

I've heard of people riding freehubs on track bikes(you still only have one gear but don't have to constantly pedal), you might want to look into this type of rear wheel, but if you're comfortable on a track bike all i can say is go for it.

I asked a fellow club member (and sanctioned USAT official) this question. There is a bike legality issue with the fixed gear involving brakes. If you have BOTH a front and rear brake (and the bike is in otherwise reasonable condition) then you would be allowed to race with it. My bike has only a front brake, and I tried to argue with him that my legs are the rear brake, but to no avail, and he believes other officials would feel as he does. As far as would you want to ride the fixie for the race? Only if it's a flat course, is what I would say.