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.


2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us