In a course in Geometry we where asked to geometrically construct the sum of two points $x$ and $y$ on a projective line by help of "the theorem on complete quadrilaterals". This theorem (as stated in our lecture notes) says that if $A,B,C,D$ are four points in general position in a real projective plane and if $P=AB\cap CD$ and $Q=AD\cap BC$, $l=PQ$, $X=l\cap BD$, $Y=l\cap AC$. Then the pair of points $\{P,Q\}$ on $l$ separates the pair $\{X,Y\}$ harmonically, meaning that the cross ratio $Cr(P,X,Q,Y)=-1$
Projective planes are assumed to satisfy desargues theorem in my course.
I have the following idea, which I first present and then I will add my concerns.
If $x=y$, then $Cr(0,x,2x,\infty )=-1$
If $y>x$ then $Cr(x, \frac{x+y}{2},y, \infty )=-1$
We use this to construct $\frac{x+y}{2}$. Let $P=x$ and $Q=y$, we let $r_1$, $r_2$ be arbitrary lines intersecting in $x$. Furthermore let $A$ and $C$ be the intersection points of $r_1$ and $r_2$ with the line $l_{\infty }$ at infinity respectively. Now draw the lines $AQ=h_1$, $CQ=h_2$. We let $D=r_2\cap h_1$ and $B=r_1\cap h_2$. Now, we are in the context of the above mentioned theorem. The line $BD$ intersects $l$ in the point $s$ such that $Cr(x, s,y, \infty )=-1$, hence $s=\frac{x+y}{2}$.
My concerns are the following:
I have problems calculating cross ratios, using Berger I would believe $Cr(x, y, \frac{x+y}{2}, \infty )=\frac{\frac{x+y}{2}-x}{\frac{x+y}{2}-y}=-1$, and in the same way I get $Cr(0,2x,x,\infty )=-1$, this seems completely unreasonable however since $2x$ is not between $0$ and $x$ which makes it impossible for these calculations to be correct, please help me clearly write down the standard expression for these crossratios, because I am somehow misunderstanding page 125 in berger $Cr[a,b,c, \infty ]=\frac{\overline{ca } }{\overline{cb}}$.
Independent on which one of the calculations of crossratio is correct, I run into problems in making the argument when the unknown quantity is on position three, i.e "when I want to construct the point $Q$".
I realized that for constructing 2x, I just map the x-axis or the y-axis to $l$ by the projective transformation taking 0 to 0 and 1 to x and infinity to infinity then the image of 2 is my desired point, hence I think that this together with previous calculations solves the problem, if $Cr(x, \frac{x+y}{2},y, \infty )=-1$ that is.
Any comments, hints or general opinions about crossratios and "geometric construction" are welcome.