This is a though one.
It seems like the locus of the centers $B$ of the desired circles all lie in a parabola. How to figure a nice simple way to construct those circles with ruler and compass?
I take it all back: the locus is not a parabola even though it looks like one from the distance.
Let $\Gamma$ and $\ell$ the initial circumference and line, respectively.
Choose any point $P\in\Gamma$. If you want the circles to be orthogonal in $P$, draw the line $\ell_1$ that joins the center of $\Gamma$ with $P$ and its perpendicular $\ell_2$ that passes through $P$.
Now, let $O=\ell_1\cap\ell$ (possibly infinity point). Note that $OP=OQ$, where $Q$ is the point where the circle you want to build will be tangent to $\ell$ (you can find $Q$ drawing a circle with center $O$ and radius $OP$).
Whit this, the perpendicular to $\ell$ through $Q$ must intersect $\ell_2$ at the center of the circle you want.