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Topic: Overlapping circles
Replies: 9   Last Post: Jun 17, 2009 2:06 AM

 Messages: [ Previous | Next ]
 Rouben Rostamian Posts: 193 Registered: 12/6/04
Re: Overlapping circles
Posted: Jun 6, 2001 6:03 PM

W.H. wrote:

> Two circles of diameter (d) and (2*d) overlap (not over the
> centres) so that they have a common chord of length (c). The
> measurement across the two circles perpendicular to the chord
> is (s) distance. I need to find the diameters in terms of (c)
> and (s).

Let's work with arbitrary diameters d1 and d2 rather than d
and 2d. Later we will substitute d1 = d and d2 = 2d.

You assume the circles intersect. This means that s < d1 + d2.

You also assume that one circle is not completely inside
the other. This means s > d1 and s > d2, which implies
that 2s > d1 + d2. Therefore s > d1/2 + d2/2.

Let O1 and O2 be the centers of the two circles. Let E be one
of their intersection points. Focus on the triangle O1-E-O2.

It's easy to see that the distance between O1 and O2 is
s - d1/2 - d2/2.

Therefore the three sides of the triangle are d1/2, d2/2 and
s - d1/2 - d2/2. We can express its area by plugging these
into Heron's formula. We get:

(1) Area^2 = s (s - d1) (s - d2) (d1 + d2 - s) / 16.

Note that the right hand side is positive due to the
inequalities stated above.

At the same time, the area can be expressed as one half of
the base times height. But the base is s - d1/2 - d2/2 and
the height is c/2. Therefore

Area = (1/2) (c/2) (s - d1/2 - d2/2)

which gives:

(2) Area^2 = [ (1/2) (c/2) (s - d1/2 - d2/2) ]^2.

Now equate the two different expressions for Area^2 in equations
(1) and (2):

(3) s (s - d1) (s - d2) (d1 + d2 - s) / 16
= [ (1/2) (c/2) (s - d1/2 - d2/2) ]^2

Equation (3) establishes a relationship between the four
quantities d1, d2, s, and c. Given any three of them, you
can calculate the remaining one.

This completes the analysis of the general case. You have
asked specifically for d1 = d and d2 = 2d. In that case,
equation (3) reduces to:

- 4 s (s - d) (s - 2d) (s - 3d) = c^2 (2s - 3d)^2 .

As you see, this is a cubic equation in d. Apply any of the
known methods to solve it.

-- Rouben Rostamian <rostamian@umbc.edu>