Projective geometry (Table of contents)

Theorems of Pappus and Pascal

Theorem 4 (Pappus) The points \(A_1\), \(A_2\), \(A_3\) belong to the line \(a\), and the points \(B_1\), \(B_2\), \(B_3\) belong to the line \(b\). Assume that \(A_1B_2\cap A_2B_1=C_3\), \(A_1B_3\cap A_3B_1=C_2\), \(A_2B_3\cap A_3B_2=C_1\). Then \(C_1\), \(C_2\), \(C_3\) are colinear.

Theorem 5 (Pascal) Assume that the points \(A_1\), \(A_2\), \(A_3\), \(B_1\), \(B_2\), \(B_3\) belong to a circle. The point in intersections of \(A_1B_2\) with \(A_2B_1\), \(A_1B_3\) with \(A_3B_1\), \(A_2B_3\) with \(A_3B_2\) lie on a line.