In a convex pentagon consider the five lines joining a vertex to the midpoint of the opposite side. Show that if four of these lines pass through a point, then so does the fifth.
Solution
Let the pentagon be ABCDE. Use vectors. Take the origin O to be the point at which the lines from each of A, B, C D to the midpoint of the opposite side meet. Take the vector OA to be a, OB to be b etc. The side opposite A is CD. Its midpoint is (c + d)2. So we have a x (c + d) = 0 and hence a x c = d x a. Similarly, b x d = e x b, c x e = a x c, d x a = b x d. Hence e x b = b x d = d x a = a x c = c x e, which shows that the line from E also passes through the origin.
© John Scholes
jscholes@kalva.demon.co.uk
11 Apr 2002