# Two Circles and a Limit

Analytic Proof

### Konhauser, Velleman, and Wagon

We are solving the Two Circles and a Limit problem:

A stationary circle of radius 3 is centered at

The problem has been included in a wonderful book *Which Way Did the Bicycle Go?* by J. Konhauser, D. Velleman, S. Wagon as Problem #5. This is where the following solution comes from. The solution is analytic, but in an elegant way. It entirely avoids the common drudgery associated with repeated applications of the Pythagorean theorem.

Let C(r) be the circle of radius r centered at the origin; S, the other circle centered at (3, 0) with radius 3.

The former has the equation x² + y² = r², the latter, (x - 3)² + y² = 3², which can be rewritten as

x² + y² = r² and

x² + y² = 6x.

Since point B(x, y) lies on C(r), x² + y² = r². And, since it lies on S,

B(r²/6, r/6·√36 - r²).

Point A has coordinates (), r), so that the slope K of line AB is

y = Kx + r.

The line intersects x-axis at (0 - r)/K = r²/(6 - √36 - r²) = (√36 - r² + 6), which when

### References

- E. J. Barbeau, M. S. Klamkin, W. O. J. Moser,
*Five Hundred Mathematical Challenges*, MAA, 1995, #396 - J. Konhauser, D. Velleman, S. Wagon,
*Which Way Did the Bicycle Go?*, MAA, 1996, #5

### Limits in Geometry

- Two Circles and a Limit
- A Geometric Limit
- Iterations in Geometry, an example
- Iterated Function Systems

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