Intersecting segments

At the intersection of two lines there are two expressions for one position vector. Use a separate scalar parameter for each line. In a non-parallel basis a, b, equal position vectors have equal coefficients. Two scalar equations therefore determine the two line parameters. Write both line positions from known endpoints; equate coefficients; solve simultaneously; substitute; finally test each segment parameter lies between zero and one before using a segment ratio. For P the midpoint of AB and M the midpoint of OA, write OX=t(a+ b)/2=u a/2+(1-u) b. Then t=u, t/2=1-u, giving t=(2)/(3) and OX:XP=2:1. Exam check: an intersection of extended lines may lie outside one or both segments. The line parameters are independent; do not assume they are equal.