Euclid's proof of proposition 2 In proposition 2 Euclid introduces points $E$ and $F$. But how do we know how to put them geometrically(without drawing)? (I thought Euclid used only shapes for proofs, but now I only wonder)
link to the proposition
Even if we knew how to continue a line, we don't know how long that line should be. Which means the line should be infinite, that is it should be a ray.
Where am I wrong?
Is Euclid's proof valid? If so, why?
 A: I do not think Euclid defines a "ray" in the same way we do, as an object that already extends infinitely far in a given direction. I think Euclid only had the ability to extend a line segment as far as you please in the direction past one end of the segment, but not to complete the extension to all points in that direction.
If Euclid had postulated the ability to construct a complete ray, he might very well have written Proposition 2 differently.
Instead of producing point $F$, he could say to construct the ray $\overrightarrow{DB}$ starting at $D$ and passing through $B$. Then $G$ is the point at which the circle with center $B$ and radius $BC$ intersects the ray $\overrightarrow{DB}$.
Likewise, $L$ is the point at which the circle with center $D$ and radius $DG$ intersects the ray $\overrightarrow{DA}$.
Since Euclid does not have the concept of a ray, but he does have the concept that you can reach any of the points you would have included in your ray,
instead of constructing an infinitely long object for the circle around $B$ to intersect he constructs a large enough part of that object.
It does not matter exactly how much of the object he constructs, as long as the far end of it (at $E$ or $F$) will be outside the circle.
And indeed this is not "proof-like" if you are accustomed to a more modern style of proof. Euclid's idea of a rigorous proof was different from Hilbert's.
A: $F$ and $E$ are named points only so that Euclid can name the lines through $DA$ and $DB$. They could be any point on those lines, but I think he wanted  points "far away" so they would lie outside the larger circle. Once you've chosen those points, $G$ and $L$ can be determined.
A: Straight lines in Euclid are what we call now segments.
This is not a proof, it's a basic construction. It is equivalent to the use of a compass to transport segments from a point to another.
$BC$ and $A$ are given. Build the equilateral triangle $ABD$. Draw the  circle with centre $B$ and radius $BC$. Draw the circle with centre $D$ and radius $DG$. Extend $DA$ to intersect the circle in $L$. $AL$ is the desired segment because it is $AL=DL-DA$ and $BC=DG-DB$ so $AL=BC$.
Don't know, to be honest, the reason why he doesn't use the compass to transport  segment $BC$ directly from $B$ to $A$.
Maybe it is because he has not yet proved that circles with equal radii are equal.
