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?

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?

Raelynn Parker · Accepted Answer

A circle has constant non-zero curvature, while the ellipse, parabola and hyperbola have changing curvature on any small section. Another &quot;conic section&quot; 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  −  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 &quot;conic sections&quot; 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.

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

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