Solving the above for r and s yields

(Ay-Cy)(Dx-Cx)-(Ax-Cx)(Dy-Cy)
r = ----------------------------- (eqn 1)
(Bx-Ax)(Dy-Cy)-(By-Ay)(Dx-Cx)

(Ay-Cy)(Bx-Ax)-(Ax-Cx)(By-Ay)
s = ----------------------------- (eqn 2)
(Bx-Ax)(Dy-Cy)-(By-Ay)(Dx-Cx)

Let P be the position vector of the intersection point, then

P=A+r(B-A) or

Px=Ax+r(Bx-Ax)
Py=Ay+r(By-Ay)

By examining the values of r & s, you can also determine some
other limiting conditions:

If 0<=r<=1 & 0<=s<=1, intersection exists
r<0 or r>1 or s<0 or s>1 line segments do not intersect

If the denominator in eqn 1 is zero, AB & CD are parallel
If the numerator in eqn 1 is also zero, AB & CD are collinear.