Pappus' Theorem with parallel lines
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The diagram above shows a special case of Pappus' theorem where the intersection points all lie on the line at infinity. The points \(A\), \(B\) and \(C\) all lie on the line \(WX\), and the points \(D\), \(E\), \(F\) all lie on the line \(YZ\). The points \(C\) and \(F\) are defined so that \(BF\) is parallel to \(AE\) and \(CE\) is parallel to \(BD\).
Pappus' theorem then guarantees that the lines \(AD\) and \(CF\) are parallel. You can see this from the picture, since the angles \(\angle WAD\) and \(\angle WCF\) are equal, therefore \(AD\) and \(CF\) are parallel by Euclid Proposition I.28.
In the diagram above, you can click and drag the points \(A\), \(B\), \(C\), \(D\) to see that the two red lines are always parallel. You can also move the line \(WX\) by clicking and dragging the points \(W\) and \(X\), and similarly for the points \(Y\) and \(Z\).
