I'm looking for a synthetic proof for the following.
Let P and Q be the circumcircle and incircle respectively of an arbitrary triangle. If ABC is a triangle inscribed in P with AB and AC tangent to Q, then BC is tangent to Q.
1. "RE: Circumcircle/Incircle Question"
In response to message #0
What do you mean by synthetic? One way or another you have to establish a certain relationship between the radii of P and Q and the distance between their centers. This is Poncelet's Porism for a 3-gon. See https://mathworld.wolfram.com/PonceletsPorism.html.
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