Two Circles and a Limit
Proof #3 by Mariano Perez de la Cruz

We are solving the Two Circles and a Limit problem:

A stationary circle of radius 3 is centered at (3, 0). Another circle of variable radius r is centered at the origin and meets the positive y-axis in point A. Let B be the common point of the two circles in the upper half-plane. Let E be the intersection of AB extended with the x-axis. What happens to E as r grows smaller and smaller?

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Let F be the midpoint of AB, D the intersection of OF with the stationary circle in the upper half-plane, C the rightmost point of the stationary circle.

D is the circumcenter of the isosceles ΔAOB. This is because angles BOD and BCD are inscribed angles subtended by the same arc, whereas ∠AOF = ∠BOF, on one hand and ∠AOF = ∠OCD, on the other. (The latter because the angles have orthogonal sides.) It follows that OD = DB. In the limit, when A, B, F tend to coalesce, D will tend to the midpoint of OF.

ΔODC is right as is the angle OFE. Therefore, EA||CD. With the previous observation, the ratio EF/CD will tend to 2, while CD will approach OC. EF will then tend to twice the segment OC.

Limits in Geometry

• Two Circles and a Limit
  • Analytic Proof
  • Barbeau, Klamkin, Moser
  • Proof #2 by Stuart Anderson
  • Mariano Perez de la Cruz
  • Trigonometric Solution
• A Geometric Limit
• Iterations in Geometry, an example
  • Iterations in Geometry, a generalization
• Iterated Function Systems

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