Parable of the Parabola
Vladimir Dragovi\'c, Mohammad Hassan Murad

TL;DR
This paper explores properties of polygons inscribed in circles and circumscribed about parabolas, proving related geometric theorems without relying on Poncelet's or Cayley's theorems, using planimetric methods.
Contribution
It provides new proofs and characterizations of polygons inscribed in circles and circumscribed about parabolas, independent of classical elliptic curve theory.
Findings
A circle contains the focus of a parabola iff a triangle inscribed in it is circumscribed about the parabola.
If the circle's center coincides with the parabola's focus, a quadrilateral inscribed and circumscribed exists and is an antiparallelogram.
For non-coincident focus and center, a specific intersection point determines the existence of such quadrilaterals.
Abstract
We study triangles and quadrilaterals which are inscribed in a circle and circumscribed about a parabola. Although these are particular cases of the celebrated Poncelet's Theorem, in this paper we {\it do not assume} the theorem but prove it along the way. Similarly, our arguments here \emph{are logically independent} from Cayley's condition, describing points of a finite order on an elliptic curve or any other use of the theory of elliptic curves. Instead, we use purely planimetric methods, including the Joachimsthal notation, to fully describe such polygons and associated circles and parabolas. We prove that a circle contains the focus of a parabola if and only if there is a triangle inscribed in the circle and circumscribed about the parabola. We prove that if the center of a circle coincides with the focus of a parabola, then there exists a quadrilateral inscribed in the circle and…
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Taxonomy
TopicsSpace Science and Extraterrestrial Life · Disaster Response and Management
