Euclidean constructions

Question

I have been playing with a Euclidean geometry application call euclidea. I have been completely stuck on this level for a while now. Any ideas how to solve this problem using only a compass and straight edge?

Answer 1

Guess and fix. First draw $AX$ and $BY$ of arbitrary but equal length:

We hope that $|XY|$ would equal $|AX|$ and $|BY|$, but alas, that is not the case.

If we construct $A'$ such that $AXYA'$ is a parallelogram, what we want is $|A'A|=|A'Y|$. But ...

... we can do that by scaling the entire red part of the diagram around $B$ until the circle hits point $A$. And that is easy using similar triangles:

Draw $AB$; it intersects the circle at $D$. Construct $AE$ parallel to $DY$. Finally (not shown) construct the last point on $AC$ equidistant from $A$ and $E$.

---

For a slightly shorter construction one can choose $Y$ to equal $C$ such that $A'$ will be on the existing line $CA$. This saves one construction-of-parallel-line. (And $X$ needs never be constructed explicitly).