What seems like a surprising fact (Desargue's Theorem) becomes intuitively obvious once you don't restrict the points to a plane. The three points **a**, **b** and **c** determine a plane and similarly for **a'**, **b'** and **c'**. The line **ab** and the line **a'b'** are confined to their respective planes and thus their intersection must lie on both of these planes. Similarly for the intersection of lines **bc** and **b'c'**, and that of **ac** and **a'c'**. So we now have the three points of intersection lying simultaneously in 2 intersecting planes. Since 2 planes intersect in a line, the intersections are collinear. Q.E.D.

Exercise for the student: Where in the above did I use the concurrency of **aa'**, **bb'**, and **cc'**?