Maybe this is a dumb question and I missed the point of something, but...
I am reading an old set of notes from when I was in college and doing some of the old homework problems. The first one I am working on defines a number, line, point or arc as constructible if you can create it using a ruler (with measurement) and compass construction.
OK.
The first problem, specifically, is to show that the point $(x,y)$ is constructible only if $(x,0)$ and $(0,y)$ are constructible.
I have an "idea" for the proof, but I am having trouble formalizing it with enough rigor.
Pf/ Assume $(x,0)$ and $(0,y)$ are constructible.
Sweep an arc of length $y$, centered at $(x,0)$.
Sweep an arc of length $x$, centered at $(0,y)$.
Their intersection is the point $(x,y)$.
$$\tag*{$\square$}$$
OK, this works really nicely when I'm the one drawing the diagram and know where $(x,y)$ is relative to $(x,0)$ and . But the truth is, when I "sweep an arc", I am really making circles centered at the two points. And those two circles will have 2, 1, or 0 intersections (though, I guess I can force 1 or 2 because of the triangle inequality). So, I guess WLOG, we can just assume there are two intersection points.
So, my question is, how can I "distinguish" between the two intersection points so that I can claim I have constructed the point $(x,y)$?