Pappus' Theorem
Back to Geometry homepage
In the picture above, the points A, B and C all lie on the line WX, and the points D, E, F all lie on the line YZ. Let P be the intersection of AE and BD, let Q be the intersection of AF and CD and let R be the intersection of BF and CE.
Pappus' theorem then guarantees that the intersection points P, Q and R are collinear, as seen from the red line in the picture.
In the diagram above, you can click and drag the points A, B, C, D, E, F to see that the intersection points P, Q, R are always collinear. You can also move the line WX by clicking and dragging the points W and X, and similarly for the points Y and Z.
In the case where P, Q and R lie on the line at infinity, Pappus' theorem says that the pairs of lines are all parallel. You can see a picture of this here.
David Joyce's webpage also has an illustration of Pappus' theorem for different geometries.
