Is there a conic tangent to each of the six lines comprising a conic-inscribed hexagon? Let $H$ be a hexagon formed by six points lying on a conic in the plane. Is there a conic tangent to each of the six lines comprising $H$?

In general, no. The necessary and sufficient condition for this to hold is that the main diagonals of hexagon are concurrent. This is called Brianchon's theorem.