Discuss an algorithm to detect whether two line segments intersect in a $2$D plane. Explain how this algorithm can be extended to identify all intersection points in a set of n line segments.

Apply your approach to find the intersections for the line segments:

L$1$: $(1,1)$ to $(4,4),$

L$2$: $(1,4)$ to $(4,1),$

L$3$: $(2,2)$ to $(5,5),$

L$4$: $(3,0)$ to $(3,5).$