Geometry : 118

Circles within a Circle

You have three circles: One has a diameter of 3 inches, another has a diameter of 2, and the third has a diameter of 1. The two smaller ones are put inside the larger one in a way so that the combined diameters of them make up the length of the largest circle's diameter. 

If you put a fourth circle in the leftover area, what is the largest area, or diameter, that this fourth circle could be? It has to touch only one point on the other three circles, but cannot cross the circles. 

I've figured out where the circle would be drawn, but can't figure out the whole thing using only math. Here is a graphic of this problem:

  

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Date: 10/14/2000 at 07:03:12
From: Doctor Floor
Subject: Re: Geometric problem.

Hi, Brian,

Thanks for writing.

When you leave off the lower halves of the circles, the remaining 
figure of semicircles is called the arbelos, or shoemaker's knife. In 
general the smaller circles can have different radii, say a and b, and 
the greatest circle then of course has radius a+b. 

This figure of the arbelos was studied by Archimedes.

The circle you are looking for is called the incircle of the arbelos.

Let O1 be the center of the circle with radius a, O2 the center of the 
circle with radius b, and O the center of the circle around them. Let 
the incircle have center C and radius r.

                               ab(a+b)
Then we will find that r = --------------
                           a^2 + ab + b^2

Proof:

Let <COO2 = t. Note that O1C = a+r, O2C = b+r and OC= a+b-r. 
Note also that OO1 = b and OO2 = a.

By the Law of Cosines in triangles OO1C and OO2C we have:

   (a+r)^2 = (a+b-r)^2 + b^2 - 2b(a+b-r) cos(180-t)
           = (a+b-r)^2 + b^2 + 2b(a+b-r) cos t........[1]

   (b+r)^2 = (a+b-r)^2 + a^2 - 2a(a+b-r) cos t........[2]

From a*[1]  + b*[2] we get:

    a(a+r)^2 + b(b+r)^2 = (a+b)(a+b-r)^2 + ab^2 + ba^2

which reduces to

    a^3 + b^3 + 2(a^2 + b^2)r + (a+b)r^2 
          = (a+b)^3 - 2(a+b)^2r +(a+b)r^2 + ab^2 + a^2b
    a^3 + b^3 + 2(a^2 + b^2)r = (a+b)^3 - 2(a+b)^2r + ab^2 + a^2b
    4(a^2+ab+b^2)r = (a+b)^3 + a^2b + ab^2 - a^3 - b^3
    4(a^2+ab+b^2)r = 4(a^2b+ab^2)

          ab(a+b)
    r = ----------
        a^2+ab+b^2