I found out that the following statement is fairly easy to prove:
Let $A$, $B$ and $C$ be thee distinct points in the plane. Let $S_{AB}$ be the circle that has the line segment $AB$ as a diameter, and similarly $S_{AC}$. Let $L_{BC}$ be the line that goes through $B$ and $C$. Then the circles $S_{AB}$ and $S_{AC}$ and the line $L_{BC}$ all go through the same point. (This happens to be the point on $L_{BC}$ closest to $A$.)
Does this result have a name? Or is there a well-known theorem about essentially the same construction? I do not recall ever encountering it, but I don't believe I'm the first one to realize it.
Note: I'm not looking for a proof of the claim. If you want to see a proof, ask me.
Lets call the point you found $X$
Via Thales' theorem (https://en.wikipedia.org/wiki/Thales%27_theorem ) you can proof that that $\angle AXB $ and $\angle AXC $ are right angles (with bases $AB$ and $AC$ )
that means that $AX$ is an altitude line.
see https://en.wikipedia.org/wiki/Altitude_%28triangle%29
so you could call point X the foot of the altitude from point A
Not sure if it has a special name. (I do doubt it)