FPG7 Pappus's Theorem Proof

We give a short proof to Pappus's Hexagon Theorem- one of the most famous and ancient results of projective geometry. Pappus proved the result using somewhat lengthy arguments based upon Euclidean geometry. We show how Pappus's Theorem follows almost immediately from the corrilaries to the fundamental theorem of projective geometry which we proved in the previous lecture. In particular, Pappus's line is precisely the axis of projectivity associated with a projection that sends the three points on one initial line, to the three points on the other initial line, and we have already discussed how this axis contains each of the three cross join points. Projective geometry is more basic and important than Euclidean geometry, because it uses less assumptions, and in concerned with statements which remain true for a much wider range of different geometric setups. In fact, with this algebraic approach, we do not even define a metric. We shall see how such ideas, as well as those of polarity, harmony and conic curves arise as natural consequences of our small set of initial axioms.