P = p*A + (1-p)*B
Q = q*C + (1-p)*D

Find p,q such that P=Q:

[ p*ax + (1-p)*bx, p*ay + (1-p)*by ] =
[ q*cx + (1-q)*dx, q*cy + (1-q)*dy ]

p*ax + (1-p)*bx = q*cx + (1-q)*dx
p*ay + (1-p)*by = q*cy + (1-q)*dy

p*(ax-bx) + bx = q*(cx-dx) + dx
p*(ay-by) + by = q*(cy-dy) + dy

( p*(ax-bx) + bx-dx ) / (cx-dx) = q
( p*(ay-by) + by-dy ) / (cy-dy) = q

  ( p*(ax-bx) + bx-dx ) / (cx-dx)
= ( p*(ay-by) + by-dy ) / (cy-dy)

  ( p*(ax-bx) + bx-dx ) * (cy-dy)
= ( p*(ay-by) + by-dy ) * (cx-dx)

  p*(ax-bx)*(cy-dy) + (bx-dx)*(cy-dy)
= p*(ay-by)*(cx-dx) + (by-dy)*(cx-dx)

p*( (ax-bx)*(cy-dy) - (ay-by)*(cx-dx) )
  = (by-dy)*(cx-dx) - (bx-dx)*(cy-dy)