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Given a triangle I draw circles around each vertex. I chose the radii of these circles so that they are all mutually tangent. There is only one way to do this. I extend these tangents. They concur at a point. Is there a name for that point?

Looking at the Encyclopedia of Triangle Centers I didn't find a match, though this has the right theme. So the 1st Ajima-Malfatti point of the outer triangle is the point-to-be-named of the inner triangle (EDIT: on closer look, I'm not sure that it is exactly). And as it happens my motivation does come from circle packing from a triangulated graph.

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Incenter. Your construction is an alternative description of the center of the inscribed circle.

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It took me a while to even realise that the tangents concurred at all, please bear with me. How do I see that this point and the incenter are the same? –  Packing point Aug 28 '11 at 10:06
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...it's equidistant from the three tangency points. I see! Thanks. –  Packing point Aug 28 '11 at 10:28
    
Also, the three excenters have the same property. In those cases there will be one circle containing the two others, so that two of the tangencies are internal and one external. –  zyx Aug 28 '11 at 18:15

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