ABCD is a convex quadrilateral. P, Q are points on the sides AD, BC respectively such that AP/PD = BQ/QC = AB/CD. Show that the angle between the lines PQ and AB equals the angle between the lines PQ and CD.
Solution
If AB is parallel to CD, then it is obvious that PQ is parallel to both. So assume AB and CD meet at O. Take O as the origin for vectors. Let e be a unit vector in the direction OA and f a unit vector in the direction OC. Take the vector OA to be ae, OB to be be, OC to be cf, and OD to be df. Then OP is ( (d - c)ae + (a - b)df)/(d - c + a - b) and OQ is ( (d - c)be + (a - b)cf)/(d - c + a - b). Hence PQ is (c - d)(a - b)(e + f)/(d - c + a - b). But e and f are unit vectors, so e+ f makes the same angle with each of them and hence PQ makes the same angle with AB and CD.
© John Scholes
jscholes@kalva.demon.co.uk
1 July 2002
Last corrected/updated 22 Oct 2002