Three tangent circles inside a larger circle

Question

Suppose you're given a circle with center $O$, I'm curious, how can one construct with ruler and compass three circles inside the larger circle such that each is tangent to the larger circle as well as to the other two?

Isn't this a sub-problem of an Apollonius problem: en.wikipedia.org/wiki/Apollonius_problem ?

Ross Millikan · Accepted Answer · 2011-02-24 16:24:52Z

up vote 0 down vote accepted

If the radius of the circle is 1, the radius of three circles that will be internally tangent is $2\sqrt{3}-3\approx 0.464$ by Soddy's formula, given in the Soddy's circles section of this. You can construct this length, then mark it off on a diameter of the circle to find one of the centers, and finish constructing the equilateral triangle.

Added: For the construction, make a right angle with 1 on one side, swing 2 as a hypotenuse, and you have $\sqrt{3}$. Then you can mark a line to get $2\sqrt{3}-3$. Draw a diameter of the circle you are given and mark off $2\sqrt{3}-3$ from the circumfrence to find the center of one circle. Construct an equilateral triangle of $2(2\sqrt{3}-3)$ bisected by the diameter and you have the three centers.

edited Feb 24 '11 at 16:24

answered Feb 24 '11 at 4:59

Ross Millikan
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	Thanks Ross Millikan, I think I should have been more explicit. Do you know of a ruler and compass construction? – user7440 Feb 24 '11 at 5:05
	Just make another line and mark off two segments of $\sqrt{3}$ end to end and back up 1 three times. This is using the modern compass that allows you to transfer a length. I have read the the Greeks used a compass that collapsed when you pick it up, but there is a proof that you can still do everything you want with one of those. – Ross Millikan Feb 24 '11 at 6:23
	Ok, so once I get a length of $2\sqrt{3}-3$, I'm thinking of inscribing an equilateral triangle in the given circle. I could then bisect each of the angles of that triangle, and then cut of points from $2\sqrt{3}-3$ from the circumference along each of those bisectors. Like you said, these will be the three centers, correct? Thank you for your time. – user7440 Feb 24 '11 at 6:34
	No, you can't inscribe the triangle in the unit circle as it is too small. It is sized so when you draw a circle of $2\sqrt{3}-3$ around each vertex they are tangent to the outer circle. As they need to be tangent to each other, the side of the triangle (correction to above) is $2(2\sqrt{3}-3)$. You really just need to construct an equilateral triangle of this side with its center matching the circle center. – Ross Millikan Feb 24 '11 at 14:40

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Three tangent circles inside a larger circle

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