Theorem without proof:
The image of a line reflected across the arc of a circle, is a circle.
The image of a ray or segment is a circular arc.

I used algebra for the proof. Having guessed (observation) that the reflection was a circle, I set up the vector problem:
There is a single point M and a corresponding constant c which satisfy, for any point X on the line,
X' = M + c (cos θ, sin θ)
The crux is this. Drop an altitude from A to the line, and draw the intersection P.
Reflect P about α:
On the reflection, P' is the closest point to the given line.
Now consider a second point Z. very far from A. The further Z gets, the closer Z' is to A:
has constant length, and r is a constant. The denominator increases without bound.
In the limit as Z gets infinitely far away, Z' = A.
So both ends of the line project to point A. The figure is closed. A line doesn't make any weird deviations, it just keeps on keepin' on. The reflection should have constant curvature. A closed figure with constant curvature is a circle, and if this is all true, AP' should be a diameter.
Test the midpoint M of AP' directly.
Success.
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The Tangent Circle Problem: