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.