In projective geometry, we know that:
For any inscribed hexagon in a conic, the intersection of the segments formed by the union of opposite vertices, are collinear.
But now, what happens if two vertices overlap, ie, if one side of the hexagon becomes tangent to the conic.
I have tried to make multiple drawings, and I see that the intersections are no longer collinear, but... do they? I know that I draw terrible, and I think that they are really colinear because the theorem never needed the points to be different.
The result still holds. Yay to projective geom.
In fact, there is an interesting 'degenerate case': $ABCD$ is a cyclic quad (or quadrilateral on a conic), then $AB \cap CD$, $BC \cap DA$, the point intersection of tangents at $A$ and $C$, and the point of intersection of tangents at $B$ and $D$ are collinear.