# Frégier and Frégier's Theorem

A curious and interesting gem is Frégier's theorem, quoted here from David Wells:

Choose any point $$P$$ on a conic, and make it the vertex of a right angle which rotates about $$P$$. Then the lines through the points of intersection, $$AA$$, $$BB$$, and so on, will all pass through a fixed point $$X$$ which lies on the normal at $$P$$, that is, on the line through $$P$$ perpendicular to the tangent at $$P$$.

Who was Frégier, and where can I find the earliest publication of his theorem?

The theorem of Frégier (Correspondance sur l'Ecole Royale Polytechnique tome III, p.394, 1816) is a limiting case of 144(a) (See footnote on p.324 of "An Introduction to the ancient and modern geometry of conics, being a geometrical treatise on the conic sections with a collection of problems and historical notes and prolegomena" by Taylor, Charles, 1840-1908).

For quick view of the above reference, scroll down to p.394, or use Contents and go to Section 11 for p.394.

Frégier appears to have published the result in

M. Frégier, Géométrie analitique. Théorèmes nouveaux sur les lignes et surfaces du second ordre. Annales de Mathématiques pures et appliquées, tome 6 (1815-1816), p. 229-241

Here's a PDF.

and what looks like the relevant theorem The following article helped me track this down, via some citations in the introduction:

LIPPS, KATHERINE SUE. “Envelopes of Certain Families of Conics.” Pi Mu Epsilon Journal, vol. 2, no. 8, 1958, pp. 359–363. JSTOR, www.jstor.org/stable/24344939. Accessed 27 June 2020.

For anybody who wants to access Gergonne's Annales, there is an index at http://www.numdam.org/item/AMPA/