Introduction to Tangent Circles and Descartes Circle Theorem

In this article, we explore the process of finding the radius of a fourth circle that is externally tangent to three given circles. This problem is classic in geometry and can be solved using Descartes Circle Theorem. This theorem provides a relationship between the curvatures of four mutually tangent circles.

Solving for the Radius of the Fourth Circle

Consider three given circles with radii r1 45, r2 45, and r3 72. We aim to find the radius r4 of the fourth circle that is externally tangent to each of these circles.

The first step is to calculate the curvatures of the given circles. The curvature ki of a circle is the reciprocal of its radius:

k1 1/r1 1/45

k2 1/r2 1/45

k3 1/r3 1/72

Let the curvature of the fourth circle be k4. Descartes Circle Theorem states that for four mutually tangent circles, the following relationship holds:

(k1 k2 k3 k4)2 2(k12 k22 k32 k42)

Step-by-Step Calculation

Substitute the curvatures of the three given circles into the equation:

(1/45 1/45 1/72 k4)2 2(1/452 1/452 1/722 k42)

Calculate k1 * k2 * k3:

1/45 * 1/45 * 1/72 (1*1*1) / (45*45*72) 1 / 212784

To add these fractions, we need a common denominator, which is 3600 (LCM of 45, 45, and 72).

Convert each term:

1/45 8/3600, 1/45 8/3600, 1/72 5/3600

Thus:

1/45 * 1/45 * 1/72 (8*8*5) / 36002 320 / 25920000 7/120

The equation now simplifies to:

(7/120 k4)2 2(1/452 1/452 1/722 k42)

Calculate 1/452 and 1/722:

1/2025 and 1/5184

The common denominator for these fractions is 103680.

Convert each term:

1/2025 51/103680, 1/5184 20/103680

Thus:

1/452 1/452 1/722 (51 51 20) / 103680 122 / 103680

The revised equation is:

(7/120 k4)2 2(122/103680 k42)

Solving this equation, we get;

k4 k1 * k2 * k3 - 2 * sqrt(k1 * k2 * k3 * (k1 k2 k3))

Substituting k1, k2, k3, we find:

k4 7/120 - 2 * sqrt(7/120 * (7/120 * 3 * 7 - 7/120))

This simplifies to:

r4 1/k4 ≈ 30

Conclusion on Tangent Circles and Descartes Circle Theorem

The process outlined in this article demonstrates how to use Descartes Circle Theorem to find the radius of a fourth circle tangent to three given circles. This method is valuable in solving geometric problems and has practical applications in fields such as engineering and architecture.

Keywords

Descartes Circle Theorem, Tangent Circles, Radius Calculation