An Introduction to Pappus' Theorem

Pappus' Theorem was discovered by Pappus of Alexandria in the 4th century AD, and has extraordinarily beautiful properties that make it one of the nicest constructions to study in projective geometry.

Begin with two lines, l and m, in the projective plane. Then choose three points on each line, label the three points on l X1, X2, and X 3  and the three points on m Y1, Y2 , and Y3. Then construct the point Z1 by intersecting the line \(\overline{X_{2} Y_{3}}\) with the line \(\overline{X_{3} Y_{2}}\) .Similarly we can construct the point Z2 as the point that intersects both the line \(\overline{X_{1} Y_{3}}\) and the line \(\overline{X_{3} Y_{1}}\) . Finally we define the point Z3 to be the intersection of the line\(\overline{X_{1} Y_{2}}\) and the line \(\overline{X_{2} Y_{1}}\) . Pappus' Theorem states that the three points we just constructed, Z 1, Z2, and Z3 are collinear.

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Above is an interactive applet that demonstrates Pappus' Theorem. The blue dots represent the six points X1 , X2, X3, Y1, Y 2, and Y3 and the red dots represent the points Z1, Z2, and Z3.You can drag the blue dots around and the red dots are forced to change whenever the blue dots move. Notice that the red dots remain collinear.

Next Step: Permutations and Pappus' Theorem