One day I was too lazy to go get a level and didn’t want to measure. So, I tried to apply projective geometry to draw one with a no-compass, straightedge-only construction.
Two different parallel lines and a point not on them are given. Using only straightedge construct a line passing through the given point that is parallel to the two given lines.
Solution
Let the two given lines be called $\alpha$ and $\beta$, and the given point $P_1$. First we draw two points $A_1$ and $A_2$ on $\alpha$ and learn how to construct the midpoint of the segment $A_1A_2$.
Constructing midpoint of a segment:
- Take a point $B_1$ on $\beta$ and draw the line $A_1B_1$.
- Take a point $C$ on $A_1B_1$, that is outside $\alpha$ and $\beta$.
- Draw the line $CA_2$ and let $B_2$ be the point of intersection of this like with $\beta$.
- Draw the lines $A_1B_2$ and $A_2B_1$. Let $D$ be the point of intersection of these lines.
- Draw the line $CD$. The point of intersection of this line and the line $\alpha$ is the midpoint of $A_1A_2$.
Construction of the third parallel line:
- Select to points $A_1$ and $A_2$ on $\alpha$.
- Construct the midpoint $A_3$, of the segment $A_1A_2$. [The line $\beta$ is no longer needed.]
- Draw the line $A_1P_1$.
- Select a point $Q$ on the line $A_1P_1$, different from $A_1$ and from $P_1$.
- Draw the lines $QA_2$ and $QA_3$.
- Draw the line $A_2P_1$ and let $R$ be its intersection with $QA_3$.
- Draw the lines $A_1R$ and $QA_3$ and their intersection $P_2$.
- The line $P_1P_2$ is parallel to the line $\alpha$.