# 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.

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