1. The problem statement, all variables and given/known data This is taken from STEP II 2003 Q5. The position vectors of the points A, B and P with respect to an origin O are ai , bj and li + mj + nk , respectively, where a, b, and n are all non-zero. The points E, F, G and H are the midpoints of OA, BP, OB and AP, respectively. Show that the lines EF and GH intersect. Let D be the point with position vector dk, where d is non-zero, and let S be the point of intersection of EF and GH. The point T is such that the mid-point of DT is S. Find the position vector of T and hence find d in terms of n if T lies in the plane OAB. 2. Relevant equations - 3. The attempt at a solution I can't seem to make any headway with this at all, some guidance would be appreciated. There's a solution given which I don't understand: I agree with E = a/2. I don't understand how F = (p+b)/2. If OB + BP = OP, then BP = OP - OB, so F = ½BP = ½(p-b), not ½(p+b)? I agree with G = b/2. I don't understand how H = (p+a)/2, for the same reason that I don't agree with F = (p+b)/2. Can anyone help? I must be doing something very wrong because I am getting that EF and GH are identical (so intersect everywhere).