I need to construct from a given triangle $\Delta ABC$ a congruent triangle $\Delta XYZ$ and I know how to do this if I'm allowed to "measure an angle with a compass" and carry that measurement over to another point--but I need to do this construction without such an ability.
Now I actually have, I think, done the construction but I cannot complete the proof. Here's what I've done so far.
For a given triangle, select a random point $X$. Below I will construct a point $Y$ so that $AB=XY$.
Draw a circle centered at $A$ passing through $X$ and a circle centered at $X$ passing through $A$, and call one of the intersections of these $D$. Form $\angle ADX$. Draw a circle centered at $A$ passing through $B$ and label the intersection with $\overrightarrow{DA}$ that is farthest from $D$ the point $E$. Draw the circle centered at $D$ passign through $E$ and label the intersection with $\overrightarrow{DX}$ the point $F$. I can prove that $XF=AB$.
I can repeat the process to get a point on $\overrightarrow{DX}$ so that the length is the same as $AC$ too. Call the point $Z$
I believe that if I form the circle centered at $X$ through $F$ and the circle centered at $F$ through $X$ and label an intersection $Y$, then $\Delta XYZ \cong \Delta ABC$. However, I can only use SSS or SAS to prove this and I only see how to prove that $XY=AB$ and $XZ=AC$. I don't have any way that I know of to prove that angles are congruent except by proving triangles are congruent and using CPCTC. That leaves proving $YZ=BC$ which I'm struggling to do.
Many thanks for any help.
If it's useful to anyone, here is a link to a Geogebra file demonstrating the construction. Since the picture is convoluted I figured sharing the file itself might be more helpful so that you can make parts of it disappear as you walk through the construction steps.
[EDIT: I decided to go ahead and manipulate things in my Geogebra construction and discovered that, actually, my construction did not construct a congruent triangle.]