Self-Conjugate Triangle needs Two Sides to be Specified

Definition
Let $\CC$ be a circle.

Let $\triangle PQR$ be a triangle such that:
 * $PR$ is the polar of $Q$
 * $QR$ is the polar of $P$

with respect to $\CC$.

Then
 * $PQ$ is the polar of $R$

and so $\triangle PQR$ is conjugate to itself.

Hence a self-conjugate triangle requires just two sides to be specified as polars of their opposite vertices.

The third side is the polar of its opposite vertex as a consequence.

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 $PQ$.

Also see

 * Definition:Conjugate Triangles
 * Definition:Self-Conjugate Triangle