10th CanMO 1978 Problem 4 ABCD is a convex quadrilateral with area 1. The lines AD, BC meet at X. The midpoints of the diagonals AC and BD are Y and Z. Find the area of the triangle XYZ.
Use vectors with origin X. Let XA = a , XB = b , XC = h b , XD = k a . Then area ABCD = area XCD - area XAB = (hk - 1) | a x b | = 1, and area XYZ = |( a + h b ) x ( b + k a )/4| = (hk - 1)/4 | a x b | = 1/4. ( b x a = - a x b , a x a = b x b = 0). 10th CanMO 1978 © John Scholes jscholes@kalva.demon.co.uk 10 June 2002