Polar is Locus of Harmonic Conjugates wrt Ellipse

Theorem

Let $\EE$ be an ellipse embedded in the plane.

Let $P$ be an arbitrary point in the plane.

Let $\LL$ be the polar of $P$ with respect to $\EE$.

Then $\LL$ is the locus of harmonic conjugates with respect to $\EE$.

Proof

This theorem requires a proof. In particular: Need to craft the definition of harmonic conjugates with respect to $\EE$. See Harmonic Property of Pole and Polar wrt Circle and explore what Sommerville says about this in chapter $\text {III}$ section $5$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $3$.