Hmm. I think best thing to do would just go in and get whatever they suggest.
You can. But if you pick a non-tri shop or a commuter/mountain bike shop, they won't really be able to suggest the "right" thing for you.
This way, if you (or the person who asked the question), understands the differences, you already have a head start. If you want to do half-ironman distances next year and they try to sell you a flat-handlebar road bike, you'll be able to say no - even if they insist it will work (and it will work, just not as well)
That said, many do find having a true road bike and a true tri bike to be an advantage for training. And don't forget the cross bike for the winter season.

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You can clip on aero bars to get the same positional effect and be able to train more easily that way.
I know you sort of corrected this in your next sentence, but the geometry really is different between the two, not just the handlebars but the entire frame. It is really important if someone wants to make the investment once and not end up with 3 bikes in the course of a few years. Again, a good shop will be able to walk you through all this and help you sort out your options.
Yes, the clip ons will get you in an aero position but it won't do anything for the rest of your body.
I do agree that many beginners can start out with a decent road bike and do many a sprint and be competitive for quite awhile this way.
I just took a look at Jamis (on par with Trek/Specialized/Giant) and their entry level tri bike is $1775 (aluminim frame). The next model up - carbon - is $2500. Really not bad.
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Since then I've moved on to a road bike but at the time I was younger, ignorant and motivated so it all worked out but I doubt you'll see me trying a century on a mountain bike again.

Lethal
Ah hell Lethal, pick a mostly-dirt century and you'll be good to go.
