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Inspired by this Sangaku-style question about a constellation of circles, I've come up with the following question.

How can we construct, with straightedge and compass, the following diagram?

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Description: In a (large) circle, two equal length chords share a point on the circle. In each of the two segments thus formed, a largest possible circle is inscribed. A third circle touches the large circle and both chords. The three inscribed circles are of equall radii.

The ratio of the large circle's radii to the smaller circles' radii is $\phi+1$, where $\phi=\frac{1+\sqrt5}{2}$ is the golden ratio (proof).

If we construct the chords, then the circles are easy to draw. But how can we construct the chords?

Dan
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1 Answers1

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The ratio of the smaller circle to the larger circle is $$\frac{1}{\phi+1}=\frac{3-\sqrt{5}}{2}$$

If you decide on the radius of the larger circle first, say $1$ unit, then draw a straight line of length $3$ units.

Take a point $2$ units from one end of the line, and construct a perpendicular from there of length $1$ unit to make a right-angled triangle whose hypotenuse is $\sqrt{5}$. Mark off the length $\sqrt{5}$ from one end of the first line. Bisect the remaining length of that line to make a length of $$\frac{3-\sqrt{5}}{2}$$

Now that you have the smaller radius you can draw the upper circle first, then the chords, and finally the two side circles, a process which should be fairly routine.

David Quinn
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