Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most interested in the case when D is a triangle or convex quadrilateral. Suppose also that $C_1$ & $C_2$ are not tangent at the same point on the boundary of D. Does anyone know of a result which states that $C_1$ & $C_2$ must intersect in 4 distinct points inside D? I am not assuming that $C_1$ & $C_2$ are convex, though I would be interested in the result for that case. I'm also not assuming any smoothness assumptions on $C_1$ & $C_2$, though again I would be interested in the result if $C_1$ & $C_2$ are analytic curves.