Its very common in illustration to want to draw a line towards a vanishing point that is off of the page.
The specific problem is this: lets say we draw two line segments on a piece of paper lying on a flat surface. The line segments are drawn such that, if extended beyond the edge of the page, they intersect at a point $A$. Next, we draw a point $B$ anywhere on the page.
Without actually extending the two line segments to find $A$, is there a way to draw a line segment such that, if extended to a line, would intersect both $A$ and $B$ using only a straight edge and compass construction?
Looking for an answer with the fewest number of steps.
Best Answer
The following solution to the problem uses the ruler only and is based on Desargues's theorem.
Let $R=AB\cap A'B'$ be the point out from the sheet and $P$ a point on the sheet. It's required to draw the line $PR$.
Let $O=AA'\cap BB'$, and choose a line $c\supset O$. Let $C=c\cap AP$ and $C'=c\cap A'P$. The triangles $ABC$ and $A'B'C'$ are homologous since $O=AA'\cap BB'\cap CC'$, thus $R=AB\cap A'B'$, $P=AC\cap A'C'$ and $Q=BC\cap B'C'$ lies on a line. Consequently, $PQ$ is the required line.