Lecture 8

Theorem (complete quadrilateral). Consider four lines with intersection points $A$, $B$, $C$, $D$, $P$, and $Q$ as shown in the picture. Then the cross-ratio of the intersection points $X$ and $Y$ of the diagonals with the line $PQ$ and $P$ and $Q$ is:

\[ \cr(P,X,Q,Y)=−1.\]

 Complete quadrilateral and quadrangular set

Proof of theorem on complete quadrilateral (multi-ratio).
Consider the multi-ratio with

$P_{1} = Q_{1} = P$, $P_{3} = Q_{3} = Q$, $Q_{2} = X$, $P_{2} = Y$,

where $p$ is the affine coordinates of $P$, $q$ the ones of $Q$ etc.

-1 &= \mathrm{m}(P, X, Q, P, Y, Q) \\
&= \frac{p-x}{x-q} \frac{q-p}{p-y} \frac{y-q}{q-p} \\
&= \frac{p-x}{x-q} \frac{q-y}{y-p} \\
&= \mathrm{cr}(P, X, Q, Y). \\


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