Arc 1 Laurence D. Finston

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Given a circular arc q, find the radius r and the angle θ subtended by q at the center M of circle k.

Solution:

Let q be the arc of a circle k with unknown center M and unknown radius r. Let p_{0} and p_{1} be the end points of q. Let p_{2} be the point halfway between p_{0} and p_{1}. Let p_{3} be the intersection of q and the perpendicular to the line p_{1}p_{0} through p_{2}. Let p_{4} be the point halfway between p_{0} and p_{3}. Let p_{5} be the intersection of q and the perpendicular to p_{0}p_{3} through p_{4}. The lines p_{3}p_{2} and p_{5}p_{4} intersect at M. Let a be the distance from p_{3} to p_{2}, i.e., the magnitude of p_{3} − p_{2}, b the distance from p_{0} to p_{2}, and c the distance from p_{0} to p_{3}. Let α be ^{}p̸_{3}p_{0}p_{2} and β be ^{}p̸_{0}p_{3}p_{2}. Then α = sin(a/c) and β = sin(b/c). Let γ be ^{}p̸_{4}Mp_{3}. Since ^{}p̸_{3}p_{4}M is a right angle, γ = α. Since the arc p_{0}p_{5} is a quarter of the arc p_{0}p_{1}, the angle θ subtended by q at M is 4γ.