Conjugate Lines are Harmonic Conjugates with respect to Tangents from Point of Intersection

Theorem

Let $\CC$ be a circle.

Let $\PP$ and $\QQ$ be conjugate lines with respect to $\CC$.

Let $\PP$ and $\QQ$ intersect at $O$.

Let $OS$ and $OT$ be the tangents to $\CC$ from $O$.

Then $\PP$ and $\QQ$ are harmonic conjugates with respect to $OS$ and $OT$.

Let $P$ and $Q$ be the poles of $\PP$ and $\QQ$ with respect to $\CC$.

Because $O$ lies on both $\PP$ and $\QQ$, its polar passes through both $P$ and $Q$.

That is, the polar of $O$ is $PQ$ itself.

The polar of $O$, by definition, is the chord of contact of $OS$ and $OT$ with $\CC$.

Hence $\map O {PQ, ST}$ forms a harmonic pencil.

That is, $\PP$ and $\QQ$ are harmonic conjugates with respect to $OS$ and $OT$.

$\blacksquare$

1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $8$. Reciprocal property of pole and polar