Finding an expression for the diameter of a circle.

Question

The figure shows two circles of radius $1$ that touch at $P$. 1]1 $T$ is a common tangent line; $C_1$ is the circle that touches $C$, $D$, and $T$; $C_2$ is the circle that touches $C$, $D$, and $C_1$; $C_3$ is the circle that touches $C$, $D$, and $C_2$. This procedure can be continued indefinitely and produces an infinite sequence of circles $\{C_n\}$. Find an expression for the diameter of $C_n$.

Answer 1

I'll work with the radii instead of the diameters. Obviously, any diameter is twice the corresponding radius. Now, note from the figure below that the triangle $OAB$ is rectangular at $A$.

Thus, $$ \overline{OA}^{\,2} + \overline{AB}^{\,2} = \overline{OB}^{\,2} $$

So, $$ (\underbrace{1}_{\overline{OA}})^2 + (\underbrace{1 - R_1}_{\overline{AB}})^2 = (\underbrace{1 + R_1}_{\overline{OB}})^2 $$ which gives you

$$ R_1 = \frac{1}{4} $$

Similarly, by making $B$ be the centre of the ever-smaller circles, you should be able to see that $$ (\underbrace{1}_{\overline{OA}})^2 + (\underbrace{1 - 2R_1 - 2R_2 - \cdots - 2R_{n-1} - R_n}_{\overline{AB}})^2 = (\underbrace{1 + R_n}_{\overline{OB}})^2 \qquad(1) $$ for $n > 1$. Now define

$$ S_{n} \equiv \sum_{k\,=\,1}^{n}R_k \qquad (n \ge 1) \qquad(2) $$

so that $(1)$ can be written as $$ 1 + (1 - 2S_{n-1} - R_n)^2 = (1 + R_n)^2 \qquad(3) $$ for $n > 1$. However, $$ (1 - 2S_{n-1} - R_n)^2 = 1 + 4S_{n-1}^2 + R_n^2 - 4S_{n-1} - 2R_n + 4S_{n-1}\,R_n $$ and $(3)$ gives $$ 1 + 1 + 4S_{n-1}^2 + R_n^2 - 4S_{n-1} - 2R_n + 4S_{n-1}\,R_n = 1 + 2R_n + R_n^2 $$ which can be simplified to $$ (1 - 2S_{n-1})^2 = 4(1 - S_{n-1})R_n $$ so

$$ R_n = \frac{1}{4}\frac{(1 - 2S_{n-1})^2}{(1 - S_{n-1})} \qquad (n > 1) \qquad (4) $$

This gives $R_n$ in terms of the radii of only the circles with indices smaller than $n$ so it's very amenable to a recursive calculation. For example, we already have $R_1 = 1/4$. Then $S_1 = 1/4$ and $R_2 = 1/12$. Then $S_2 = 1/3$ and $R_3 = 1/24$. Then $S_3 = 3/8$ and $R_4 = 1/40$, and so on.

Computing a few more values in succession, it's easy to discern the pattern:

$$ S_n = \frac{n}{2(n+1)} \qquad\mbox{and}\qquad R_n = \frac{1}{2n(n+1)} \qquad (n \ge 1) $$

That these satisfy $(2)$ and $(4)$ can be easily proven by induction on $n$.

Answer 2

here is another way to look at this problem. we will use inversion on a circle.

let me set up the coordinate system. we will choose the coordinate system so that the initial circles have centers at $(1, 0), (-0,1)$ and the horizontal line touching these circle and $C_1$ is $y = 1.$ the mirror circle is the unit circle $x^2 + y^2 = 1.$ a point $(0, y)$ on the $y$-axis gets reflected to $(0,1/y).$

the images of the initial two circles are the parallel lines $y = \pm 1/2.$ the image of the circle $C_1$ has a diameter the line connecting $(0,1), (0,2).$ in the same way the image of the circle $C_n$ has a diameter the line connecting $(0,n), (0,n+1).$

reflecting these points, we find that $C_1$ has a diameter connecting the points $(0,1), (0, 1/2)$ and therefore the length of the diameter is $1-1/2 = 1/2$ and the diameter of $C_n$ is $\frac1n - \frac1{n+1} = \frac1{n(n+1)}.$

Answer 3

Notice, the radius say $r_1$of circle $C_1$ is given by the formula (see derivation) $$\color{red}{r_1=\frac{ab}{(\sqrt a+\sqrt b)^2}}$$ setting radii $a=b=1$ of larger identical circles, we get $$r_1=\frac{1\cdot1}{(1+1)^2}=\frac{1}{4}$$

Now, the radius $r_2$ of circle $C_2$ externally touching the circles $C$, $D$ & $C_1$ is given by the standard formula (see derivation) $$\color{red}{r_2=\frac{abc}{2\sqrt{abc(a+b+c)}+(ab+bc+ca)}}$$ Setting the values of $a=1, b=1, c=r_1=\frac{1}{4}$, we get radius of circle $C_2$ as follows $$r_2=\frac{1\cdot1\cdot\frac{1}{4}}{2\sqrt{1\cdot1\cdot\frac{1}{4}\left(1+1+\frac{1}{4}\right)}+1\cdot1+1\cdot \frac{1}{4}+\frac{1}{4}\cdot 1}=\frac{1}{12}$$

Similarly, the radius $r_3$ of circle $C_3$ externally touching the circles $C$, $D$ & $C_2$ is calculated by setting the values of $a=1, b=1, c=r_2=\frac{1}{12}$ in the above standard formula as follows $$r_3=\frac{1\cdot1\cdot\frac{1}{12}}{2\sqrt{1\cdot1\cdot\frac{1}{12}\left(1+1+\frac{1}{12}\right)}+1\cdot1+1\cdot \frac{1}{12}+\frac{1}{12}\cdot 1}=\frac{1}{24}$$ Thus, continuing the same procedure, we can calculate radius of any circle $C_n$ using above standard formula.