Finding ratio in which a point divides a line?
Hi! here's the question, its quite long.....
"Let ABCD be a plane quadrilaterial. Suppose that the diagonals AC and BD intersect at the point P and the sides AB and CD (extended) meet at the point Q. Let a, b, c and d be the position vectors of A, B, C and D relative to some origin O.
a) i've done the first part: if P divides AC in the ratio: alpha : (1-alpha) and BD in the ratio beta : (1-beta) then (1-alpha)a + alpha(c) = (1-beta)b + beta(d)
But now for this:
Suppose that the scalars alpha and beta in part (a) are both positive and not equal to each other. Show that (1-alpha)(a)/(beta-alpha) - (1-beta)(b)/(beta-alpha) = -(1-beta)(b)/(beta-alpha) + (beta)(d)/(beta-alpha)
and hence determine the ratios in which Q divides AB and CD (externally)."
I just don't understand how to relate the ratio division of P to Q.
Thanks in advance! =)