Triangle Conjugacy is Mutual

Theorem
Let $\CC$ be a circle.

Let $\triangle PQR$ be a triangle.

Let $\triangle P'Q'R'$ be such that:
 * $P'$ is the pole of $QR$
 * $Q'$ is the pole of $PR$
 * $R'$ is the pole of $PQ$

with respect to $\CC$.

Then:
 * $P$ is the pole of $Q'R'$
 * $Q$ is the pole of $P'R'$
 * $R$ is the pole of $P'Q'$

with respect to $\CC$.

That is, $\triangle PQR$ and $\triangle P'Q'R'$ are conjugate triangles with respect to $\CC$.

Proof
We have that:
 * the polar of $P'$ is $QR$
 * the polar of $Q'$ is $PR$

and so both polars pass through $R$.

Therefore:
 * the polar of $R$ is $P'Q'$.

Similarly:
 * the polar of $P$ is $Q'R'$

and:
 * the polar of $Q$ is $P'R'$.