### 38th Putnam 1977

**Problem B2**

P is a plane containing a convex quadrilateral ABCD. X is a point not in P. Find points A', B', C', D' on the lines XA, XB, XC, XD respectively so that A'B'C'D' is a parallelogram.

**Solution**

Let X' be the intersection of AC and BD. Take A', C' so that XA'X'C' is a parallelogram. Similarly take B', D' so that XB'X'D' is a parallelogram. Then both A'C' and B'D' have their midpoint at the midpoint of XX'. Hence A'B'C'D' is a parallelogram.

38th Putnam 1977

© John Scholes

jscholes@kalva.demon.co.uk

30 Nov 1999