Intersection Point of Two Line Segments
R, 95 bytes



function(m,a=c(1,3),r=range){plot(0,,"n",r(m[a,]),r(m[-a,]));segments(m[1,],m[2,],m[3,],m[4,])}


Try it online!

Here's a graphical solution as an appetizer while I'm working on a full-blown solution.
TIO link will not plot anything.