Two Circles and a Limit
We are solving the Two Circles and a Limit problem:
A stationary circle of radius 3 is centered at
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The use of trigonometry was suggested by the NY math teacher Patrick Honner.
Let α = ∠AOB, with O standing for the origin. Then B = (r sinα, r cosα), so that the slope of AB is given by
(r cosα - r) / (r sinα) = - sin(α/2)/cos(α/2) = -tan(α/2) = tan(180° - α/2).
This means that ∠AEO = α/2. In the right ΔAOE, EO = r / tan(α/2).
But r is a function of α - r = r(α) - which can be determined from the stationary circle. Join B to F, the second end of the diameter. ΔOBF is right,
Of course we also have
EO = r / tan(α/2) = 6 sin(α) / tan(α/2) = 12 cos²(α/2).
Clearly α and r tend to 0 simultaneously. So, as r tends to 0, cos²(α/2) tends to 1, while EO has the limit of 12.
Patrick Honner came up with a different opening for the proof.
Let X be the center of the stationary circle. ΔABX is isosceles and
Limits in Geometry
- Two Circles and a Limit
- A Geometric Limit
- Iterations in Geometry, an example
- Iterated Function Systems
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