There is an elegant geometric proof due to H.-P. Schröcker (JGG vol. 22/1) using a perspective collineation of the conic to a circle. There Frégier‘s theorem occurs as the classical Thales Theorem!
We present two proofs of Frégier's Theorem, both having their own merits. The first proof shows how to derive Frégier's Theorem from Thales' Theorem by means of a homology to a circle (Figure 1).
Figure 1. Proof of Frégier's Theorem by homology to circle
First proof
Take an arbitrary circle $𝒟$, tangent to $𝒞$ at $p$. There exist a homology $\eta$ with center $p$ that maps $𝒟$ to $𝒞$. (Its axis $A$ is the Desargues axis of two triangles that correspond in $\eta$ and are inscribed into $𝒟$ and $𝒞$, respectively.) By Thales' Theorem, the Frégier point is then $f = \eta(m)$ where $m$ is the circle center.
Second proof
For a right triangle inscribed into $𝒞$ and with right angle at $p$, denote the other vertices by $q$ and $r$. The map $\varphi\colon 𝒞 \to 𝒞$, $q \mapsto r$ (with appropriate conventions if $p$ coincides with $q$ or $r$) projects to the orthogonal involution in the line bundle around $p$. Hence, it is an involution in $𝒞$ and there exists a point, the Frégier point $f$, which is collinear with all pairs of corresponding points [1, Theorem 8.2.8]
isee
Posted 2022-11-20 22:25
