I am looking at constructible points in abstract algebra, particularly in $\mathbb{C}$. Alongside a proof of a theorem, I came across this expression which I cannot work out how it's been derived. It just comes out as follows, but some notations;
$L(z_1,z_2)$ represents the straight line that connects points $z_1,z_2 \in \mathbb{C}$. $\mathbb{Q}^{Py}$ represents the Pyhtagorean Closure of $\mathbb{Q}$.
The theorem is stated as follows
A point $z \in \mathbb{C}$ is constructible if and only if $z \in \mathbb{Q}^{Py}$.
Omitting the bits I understood(The proof uses induction), the specific part I don't understand is,
Say $z$ lies on a "line meeting another line", namely $\{z\} \in L(z_1,z_2) \cap L(z_3,z_4)$ where the lines are distinct. Then $\exists a,b \in \mathbb{R}$ such that
$$z=az_1+(1-a)z_2$$
$$z=bz_3+(1-b)z_4$$
Well, in all honesty, it looks familiar; points on a line that goes through $2$ distinct points. I think the above expressions come from rather elementary facts and ideas but as much as I am embarrassed to say this, I can't see how it gets derived. Considering the Complex plane as $\mathbb{R}^2$, taking $x,y$ coordinates, I thought I could find $a$ or $b$ wrp to $z_j=x_j+iy_j$ but I cannot.
Would someone show me how that part is derived?
You have the right idea, you should be considering it as $\mathbb{R}^2$. The line $L(z_1,z_2)$ can be defined as the image of the straight-line path between its endpoints:
$$ f : [0,1] \rightarrow \mathbb{C} \\ f(t) = z_1 + t(z_2 - z_1) \\ = tz_2 + (1-t)z_1 $$
So any point on the line can be represented as $f(t)$ for some $t$. The step you're looking at is just choosing $a$ and $b$ to be the appropriate values for the point being considered, in terms of the two different lines it is on.