Constructing Points Given that you have two lines intersecting at the origin "0", with the unit "1" marked on each line, and "2" marked on the second line, clearly show how you would construct the point (2+1), the point 2*2=4, and the point $\frac{1}{2}$ on the second line, using the appropriate theorems (Desargues, Pappus, the Scissors Theorem).
 A: The parallel to the line $\color{red} 1\color{blue} 1$ throug $\color{red}2$ gives you $\color{blue}2$. Then the parallel to $\color{red}2\color{blue}1$ through $\color{blue}2$ gives you $\color{red}3$.
The parallel to $\color{red}3\color{blue}1$ through $\color{blue}2$ gives you $\color{red}4$.
The parallel to $\color{red}1\color{blue}2$ through $\color{blue}1$ gives you $\color{red}{\frac12}$.
To be honest, this $3$ is not really $2+1$ but rather $2+(2-1)$, and this $4$ is not $2\times 2$, but rather $2+(2-1)+(2-1)$.
A: Desargues and Pappus theorems appear usually in projective geometry where no parallel lines are available for the construction.  So you should state whether you mean Euclidean, affine or projective notion of constructing points.  The problem most naturally belongs to affine geometry with parallel line as one of the construction operations, can be done in Euclidean by implementing the affine construction using straight line and compass, and looks impossible in projective geometry.
