Inverse of Straight Line Not Through Inversion Center is Circle Through Inversion Center

Theorem

Let $C$ be a circle on center $O$ in the plane.

Let $T : X \to Y$ be an inversive transformation with $C$ as the inversion circle.

Then $O$ is the inversion center.

Let $L$ be an arbitrary straight line not containing $O$.

Let $P$ be an arbitrary point on $L$.

Let $P'$ be the image of $P$ under $T$.

Then $P'$ lies on a circle through $O$.

Proof

By Inverse of Circle Through Inversion Center is Straight Line Not Through Inversion Center:

the image under $T$ of a circle through $C$ is a straight line not through $C$.

By definition of inversive transformation, $T$ is an involution.

Thus, the image under $T$ of a straight line not through $C$ is a circle through $C$.

$\blacksquare$

Sources

• 1996: Richard Courant, Herbert Robbins and Ian Stewart: What is Mathematics? (2nd ed.): Chapter $\text{III}$ / $\text{II}$ Section $4$: "Geometrical Transformations. Inversion."