(オリジナルポスト)
sympyを使って(intersection)

交点が線分の間にあれば、交差です。
from sympy import *
def myKyoriHantei(ax,ay,bx,by,cx,cy,dx,dy,kx,ky):
    ans = "Nil"
    Lab=sqrt((ax-bx)**2+(ay-by)**2)
    Lcd=sqrt((cx-dx)**2+(cy-dy)**2)
    if (sqrt((kx - ax) ** 2 + (ky - ay) ** 2) <= Lab and \
        sqrt((kx - bx) ** 2 + (ky - by) ** 2) <= Lab and \
        sqrt((kx - cx) ** 2 + (ky - cy) ** 2) <= Lcd and \
        sqrt((kx - dx) ** 2 + (ky - dy) ** 2) <= Lcd
        ):
     ans="T"
    return ans
def myKousaHantei(lineSeg):
    (myAx, myAy), (myBx, myBy), (myCx, myCy), (myDx, myDy) = lineSeg
    PA, PB = Point(myAx, myAy), Point(myBx, myBy)
    SAB = Segment(PA, PB)
    PC, PD = Point(myCx, myCy), Point(myDx, myDy)
    SCD = Segment(PC, PD)
    ans=SAB.intersection(SCD)
    if(ans==[]):
        return "Nil"
    myKx=SAB.intersection(SCD)[0][0]
    myKy=SAB.intersection(SCD)[0][1]
    return myKyoriHantei(myAx, myAy, myBx, myBy, myCx, myCy, myDx, myDy, myKx, myKy)
# 例1
lineSeg=((2,3),(11,12),(10, 5),(3,11))
print('# 例1 (lineSeg:IsIntersect) ->',lineSeg,'->',myKousaHantei(lineSeg))
# 例2
lineSeg=((2,3),( 5,12),(10, 5),( 8,11))
print('# 例2 (lineSeg:IsIntersect) ->',lineSeg,'->',myKousaHantei(lineSeg))
# 例3
lineSeg=((2,3),( 5,12),(10, 5),( 6, 8))
print('# 例3 (lineSeg:IsIntersect) ->',lineSeg,'->',myKousaHantei(lineSeg))
# 例4
lineSeg=((2,3),(11,12),(10, 5),( 8, 9))
print('# 例4 (lineSeg:IsIntersect) ->',lineSeg,'->',myKousaHantei(lineSeg))
# 例5
lineSeg=((2,3),( 4, 9),(10, 5),( 5,12))
print('# 例5 (lineSeg:IsIntersect) ->',lineSeg,'->',myKousaHantei(lineSeg))
# 例6
lineSeg=((2,3),( 7,12),(10, 5),( 7,12))
print('# 例6 (lineSeg:IsIntersect) ->',lineSeg,'->',myKousaHantei(lineSeg))
# 例1 (lineSeg:IsIntersect) -> ((2, 3), (11, 12), (10, 5), (3, 11)) -> T
# 例2 (lineSeg:IsIntersect) -> ((2, 3), (5, 12), (10, 5), (8, 11)) -> Nil
# 例3 (lineSeg:IsIntersect) -> ((2, 3), (5, 12), (10, 5), (6, 8)) -> Nil
# 例4 (lineSeg:IsIntersect) -> ((2, 3), (11, 12), (10, 5), (8, 9)) -> T
# 例5 (lineSeg:IsIntersect) -> ((2, 3), (4, 9), (10, 5), (5, 12)) -> Nil
# 例6 (lineSeg:IsIntersect) -> ((2, 3), (7, 12), (10, 5), (7, 12)) -> T
sympyを使って(solve)

質問中