ABC is a triangle and P an interior point. Show that we cannot find a piecewise linear path K = K1K2 ... Kn (where each KiKi+1 is a straight line segment) such that: (1) none of the Ki do not lie on any of the lines AB, BC, CA, AP, BP, CP; (2) none of the points A, B, C, P lie on K; (3) K crosses each of AB, BC, CA, AP, BP, CP just once; (4) K does not cross itself.
Solution
Each time K crosses the boundary of a triangle it moves from the outside to the inside or vice versa. K has two endpoints, so we can find one of the three triangles ABP, BCP, CAP in which it does not start or finish. But that is impossible - on the first crossing it must go from outside to inside, on the second from inside to outside and on the third from outide to inside.
© John Scholes
jscholes@kalva.demon.co.uk
5 Mar 2002