You don't need Xlib for this. Let the two segments be
A1 = (x1, y1) -> B1 = (x1 + dx1, y1 + dy1) and
A2 = (x2, y2) -> B2 = (x2 + dx2, y2 + dy2).
Let
vp = dx1 * dy2 - dx2 * dy1
If vp == 0 the segments are parallel and there is no intersection.
Otherwise, let v = (vx, vy) be the vector between A1 and A2
vx = x2 - x1
vy = y2 - y1
Compute
k1 = (vx * dy2 - vy * dx2) / vp
k2 = (vx * dy1 - vy * dx1) / vp
If either k1 or k2 fall outside the [0, 1] interval, the segments don't intersect (but the underlying lines do intersect). Otherwise, the intersection is at
(x1 + k1 * dx1, y1 + k1 * dy1)
which incidentally, if you wonder about symmetry, will be the same point as
(x2 + k2 * dx2, y2 + k2 * dy2)
These formulas are basically similar to the answer on How do you detect where two line segments intersect? except coding from there would not necessarily be trivial for either a newbie or someone in a rush (like I have been myself many times).