Circlegeometric

Circlegeometric is the branch of geometry that concentrates on circles and the relationships between circles and lines, points, and other geometric objects. It studies the defining elements of a circle—the center, radius, diameter, and circumference—and the features formed by chords, tangents, secants, arcs, and angles subtended by arcs.

Fundamental results in circlegeometric include the inscribed angle theorem, which relates an angle formed by chords to half the measure of its intercepted arc, and its corollaries for central angles. The tangent-secant theorem describes relations between tangent and secant segments from an external point, while the power of a point gives a consistent product relation for intersecting lines with a circle. In cyclic figures, properties such as Thales’ theorem apply to right angles subtended by a diameter. Analytically, circles are described by coordinates with the equation (x − h)^2 + (y − k)^2 = r^2, enabling computation of intersections with lines and other circles.

Circlegeometric methods underpin many applications, including precise arc design in architecture, computer graphics rendering, robotic path planning, surveying, and various engineering problems that involve circular motion, gear design, or wheel geometry.

Historically, circle geometry stretches from classical Greek geometry through the work of Apollonius, with later development in analytic geometry by Descartes and in modern computational geometry for CAD and GIS.