Supposing $n$ circles on the real (affine) plane are given with a point $P$ that is not inside any of the circles.
Is there an algebraic plane curve $\mathcal C$ (i.e. pts of the curve satisfy the zero set of a bivariate polynomial) that is tangent to all $n$ circles and pass through the point $P$. Here by tangent I mean that the curve will have a common tangent to each of the circle and will not intersect any of the circles transversely (at least not in the real plane).
If 1. is true, then is there an easy construction for this curve (preferably with least degree, i.e. the total degree of the polynomial defining the curve is minimum).
I feel that we can even construct a Bezier curve (i.e. parametrizable) that is tangent to all these circles but I do not yet have a clue as to how. Any ideas would be appreciated!