It is common to extend the Euclidean plane with points at infinity, one for each direction. The the theorems of Euclidean geometry generalize amazingly well to this extended plane.
The Power of a Point Theorem states:
Given a circle and an arbitrary point $X$, consider any line which intersects the circle at points $A, B$. Then the quantity $XA \cdot XB$ is constant for any such line.
Does the Power of a Point Theorem apply to a point at infinity? How?
By point at infinity, I mean one the projective plane such that for each direction of parallel lines, there exists exactly one point at infinity.
I would think it could not. However, when examining How do we know chords joining secants are themselves parallel? (Hadamard) , I noticed that as either of the lines are rotated, and $X$ moves further away, the proof continues to work, until the lines become parallel and $X$ becomes the point at infinity:
At this point, the claim is still true, but there is no $X$ on the Euclidean Plane to apply PoP to. Which begs the question: Does PoP somehow apply to a point at infinity? Can we use it to produce any non-trivial consequences? For example, can we use it to prove:
If two circles $O,P$ intersect at points $A,B$ and two parallel lines $a,b$ pass through $A$ and $B$ respectively, and further intersect the circles at $A_o,B_o$ and $A_p, B_p$ respectively, then the chord on $O$ from $a$ to $b$ is parallel to the chord on $P$ from $a$ to $b$?
If $a,b$ intersect, this is an immediate consequence of the Power of the Poit. Is there an analog for a point at infinity that is strong enough to prove the case when $a,b$ are parallel?