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### Distance Between Points of Tangency

```
Date: 03/20/99 at 17:50:00
From: Julian Wake
Subject: Distance between contact points of circles of differing
diameters

I homeschool my daughter and have come across an interesting problem.

Let's say we have two circles that are each 5mm in diameter, and the
centers of the circles are exactly 50mm apart. If we place these
circles so that their outside edges contact the outside edge of
a 2,000mm diameter circle, the points where the smaller circles touch
the larger circle would be very close to 50mm apart, because of the
size of the larger circle.

Now if we take the same smaller circles (with their centers still 50mm
apart) and place them on the outer edge of a circle with a diameter of
only 100mm, the contact points would be much closer together because
the smaller circles would touch the 100mm circle on their "inner"
sides. How can we calculate the distance between the contact points of
the two smaller circles on the larger circle? We know it has to do
with the radii of the three circles and the length of the arc on the
larger circle, but haven't been able to come up with a formula to find
this distance.

Any insight you can offer would be much appreciated!

- Julian
```

```
Date: 03/23/99 at 12:02:49
From: Doctor Peterson
Subject: Re: Distance between contact points of circles of differing
diameters

Hi, Julian. This is an interesting problem, and actually surprisingly
easy to solve!

Let's draw the three circles:

*****           +++++++++++           *****
*     *     +++++     d     +++++     *     *
*   +---* +++---------------------++++*---+   *
*   r\*++----------------------------+*/r   *
***++ \              x              / ++***
+     \                         /     +
+        \                     /        +
+           \                 /           +
+           R  \             / R            +
+                \         /                +
+                   \     /                   +
+                      +                      +
+                                             +
+                                           +
+                                           +
+                                         +
+                                       +
+                                     +
++                                 ++
+++                            ++
+++                     ++++
+++++           +++++
+++++++++++

I'm calling the radius of the small circles r, the radius of the large
circle R, and the distance between the centers of the small circles d.
We're looking for x, the distance between the points of tangency.

The first thing to notice is that the points of tangency are on the
lines connecting the centers of the small circles to the center of the
large circle. (If that isn't obvious to you, take some time to think
about how to prove it.) We therefore have a pair of similar isosceles
triangles:

d
+-----------------------------------+
r\                               /r
+---------------------------+
\           x           /
\                   /
\               /
R  \           / R
\       /
\   /
+

We can set up a proportion between the two triangles:

x      d
--- = -----
R    R + r

From this you can solve for x in terms of R, r, and d:

R * d
x = -----
R + r

Let's play with this, as you did with the picture. Suppose R is much
larger than r; then R + r is practically the same as R, and x becomes
close to d, as you pointed out. Suppose instead that R = d/2 - r, so
that the large circle fits exactly between the small circles. Then
x = d - 2r (= 2R), as we would expect.

Did you notice that this hardly involves the circles at all? We only
had to use triangles, and there's no arc length or trigonometry
involved in the answer.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Conic Sections/Circles
Middle School Geometry
Middle School Triangles and Other Polygons

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