Segment lengths in circles pdf

Download Free PDF. Circle Chains Inside a Circular Segment Giovanni Lucca. A short summary of this paper. Download Download PDF. Translate PDF. We consider a generic circles chain that can be drawn inside a circu- lar segment and we show some geometric properties related to the chain itself. We also give recursive and non recursive formulas for calculating the centers coordinates and the radius of the circles. Point O is the intersection between the diameter and the chord.

Inside the circular segment bounded by the chord GH and the arc GBH, it is possible to construct a doubly infinite chain of circles each tangent to the chord, and to its two immediate neighbors. We set up a cartesian coordinate system with origin at O.

Beginning with a circle with center X0 , Y0 and radius r0 tangent to the chord GH and the arc GBH, we construct a doubly infinite chain of tangent circles, with centers Xi , Yi and radius ri for integer values of i, positive and negative.

Publication Date: August 31, Communicating Editor: Paul Yiu. The author thanks Professor Paul Yiu for his help in improving this paper. Lucca 2. Some geometric properties of the chain We first demonstrate some basic properties of the doubly infinite chain of circles. Proposition 1.

The centers of the circles lie on the parabola with axis along AB, focus at C, and vertex the midpoint of OB. Therefore, we can solve for unknown lengths by multiplying the external part FP by the entire secant length GP and set it equal to the product of the external part EP and the entire secant length FP of the second secant.

Thirdly, if a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment. We could also use the geometric mean to find the length of the secant segment and the length of the tangent segment, as Math Bits Notebook accurately states.

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