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 $PQ$.

$\blacksquare$

Also see

• Definition:Conjugate Triangles
• Definition:Self-Conjugate Triangle

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $9$. Conjugate triangles