An Introduction to Pappus' Theorem



	An Introduction to Pappus' Theorem Pappus' Theorem was discovered by Pappus of Alexandria in the 4th century AD, and has extraordinarily beautiful properties that makeit one of the nicest constructions to study in projective geometry. Begin with two lines, l and m, in the projectiveplane. Then choose three points on each line, label the three points on l X₁, X₂, and X₃ andthe three points on m Y₁, Y₂ , and Y₃. Then construct the point Z₁ by intersectingthe 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, Z₁, Z₂, and Z₃ are collinear. collinear understands the intersection of tag but isn't running geometry., geometry. reason." Your browser is completely ignoring the <APPLET>tag!Get Netscape. Above is an interactive applet that demonstrates Pappus' Theorem. The blue dots represent the six points X₁ , X₂, X₃, Y₁, Y₂, 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 wheneverthe blue dots move. Notice that the red dots remain collinear. Next Step: Permutations andPappus' Theorem