How to prove this theorem rhetorically?

Question

It is not possible for a part of any of three conic sections to be an arc of a circle.

It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever (algebra). But, to me, this seems to be quite a difficult task. How can this be done?

Answer 1

A circle has constant non-zero curvature, while the ellipse, parabola and hyperbola have changing curvature on any small section. Another "conic section" is a pair of intersecting lines, which has zero curvature.

Some sources also consider a pair of parallel lines to be a conic section, since that can result from a quadratic equation in two variables, such as $x^2+0y^2=1$ or $(x-y)^2=1$, even though they cannot result from the intersection of a double-cone with a plane. These also have zero curvature.

The last "conic sections" are a point and the empty set, which can be considered to be an arc of a circle with zero central angle. Your question should be more clear in leaving out the degenerate conic sections.

Answer 2

Any arc of a conic, no matter how small, determines the conic completely because a conic is given by a quadratic equation, whose full form contains 6 coefficients. Therefore, any 5 points on a conic determine it, because you can normalize one of the coefficients.