On this site it says, that we can make a conic with five tangents. I know how to construct a conic with five points, so I am wondering, how do we determine which point from tangents do we use. I know, that we cannot take three collinear points (so we cannot choose the intersections), but other than that, what are our restraints?
Use Brianchon's theorem as in the following diagram:
If $F$ is the point of contact of the tangent $CD$, then by Brianchon's theorem the three diagonals of the hexagon $ABCFDE$ are concurrent at a point $O$.
So we can intersect $BD$ and $CE$ to find $O$, and the contact point $F$ lies on the line $AO$. Similarly we can find the four other contact points.
This gives us five points, from which we can construct the conic.