Three Equal Circles in a Semicircle

A chain of three equal circles is inscribed in a semicircle, as shown below. The ratio of the radius of the semicircle to the diameter of the small circles is the golden ratio φ.
three equal circles in a semicircle

Proof

References

  1. C. A. Pickover, A Passion for Mathematics, John Wiley & Sons, 2005, p. 82

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Copyright © 1996-2017 Alexander Bogomolny

A chain of three equal circles is inscribed in a semicircle, as shown below. The ratio of the radius of the semicircle to the diameter of the small circles is the golden ratio φ.
three equal circles in a semicircle

Proof

Join the centers of three circles, as shown, and extend one line to the point of tangency of two circles.

three equal circles in a semicircle

The proof needs just one application of the Pythagorean theorem. Let r and R be the radii of the small circles and the semicircle. In the right triangle in the diagram, one of the legs equals r, the other to 2r, and the hypotenuse to R - r, giving a relation:

r² + (2r)² = (R - r)²,

which translates into

R² - 2Rr - 4r² = 0.

Solving for R/r, we get

R / r = 1 ±5.

As the minus sign would produce a negative ratio, we settle for the sign plus:

R / 2r = (1 ±5) / 2 = φ.

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Copyright © 1996-2017 Alexander Bogomolny

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