Exploring the Relationship Between Tangents and Intercepted Arcs in Conic Sections — Specifically Circles: A Supplementary Theorem

Abstract

This paper presents a novel geometric theorem demonstrating that the angle formed by two tangents intersecting outside a circle is always supplementary to the closest intercepted arc between their points of tangency. This discovery introduces a direct and efficient method for determining arc measures, offering significant improvements over traditional multi-step approaches. By leveraging fundamental properties unique to circles—namely, that tangents are perpendicular to radii at points of tangency and that central angles equal their intercepted arcs—we derive a streamlined two-step process for calculating arc measures from external angles. This contrasts with older methods requiring multiple arc calculations and geometric constructions, which are more time-consuming and error-prone. Our theorem enhances clarity and accuracy in geometric problem-solving and is supported by a visual model showing a consistent inverse linear relationship between the external angle and the intercepted arc. Furthermore, the theorem’s reliance on circular symmetry explains why it does not generalize to other conic sections such as ellipses, parabolas, or hyperbolas.

Keywords: Euclidean geometry, circle theorems, tangent lines, intercepted arcs, supplementary angles, conic sections, geometric problem-solving

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