$C, C_1, C_2, C_3$ are three circles of radii $5, 3, 2, x$ units respectively. $C_1$ and $C_2$ touch each other externally and $C$ internally. $C_3$ touches $C_1$ and $C_2$ externally and $C$ internally.

Find the value of $x$.

I've figured out that the diagram looks like this. And I have a hunch that the centre of $C_3$ will be directly above the intersection point of $C_1$ and $C_2$. I've also tried constructing right-angled triangles and applying the pythagoras theorem.
enter image description here

  • $\begingroup$ Lookup Descartes' Theorem for the general case. $\endgroup$ – dxiv Mar 12 '17 at 17:35
  • $\begingroup$ The center $M$ of $C_3$ is at distance $5-x$ from the center of $C$ hence $M=(5+(5-x)\cos t,(5-x)\sin t)$ for some $t$. The distance from $M$ to the center $(3,0)$ of $C_1$ is $3+x$ and the distance from $M$ to the center $(8,0)$ of $C_2$ is $2+x$, hence $$(5+(5-x)\cos t-3)^2+((5-x)\sin t)^2=(3+x)^2$$ and $$(5+(5-x)\cos t-8)^2+((5-x)\sin t)^2=(2+x)^2$$ Can you finish this? $\endgroup$ – Did Mar 12 '17 at 17:46

(see picture below)

Let us denote, on homogeneity grounds, $C_0=C$ and by $C_{ij}$ the contact point of circle $C_i$ with circle $C_j$.

A common feature is that points $C_i,C_j$ and $C_{ij}$ are aligned. More precisely, we are interested by the following alignements:

  • $C_1,C_{13},C_3$ are aligned in this order, thus $C_1C_3=3+x$ (external contact).

  • $C_2,C_{23},C_3$ are aligned in this order, thus $C_2C_3=2+x$ (external contact).

  • $C_0,C_3,C_{03}$ are aligned in this order, thus $C_2C_3=5-x$ (internal contact).

Let $(x_3,y_3)$ be the coordinates of $C_3$.

We can write the following system of 3 equations with 3 unknowns $x,x_3,y_3$:

$$\begin{cases}\text{square of length} \ C_1C_3: & \ \ (x_3-3)^2+y_3^2&=&(3+x)^2 & \ \ (a)\\\text{square of length} \ C_2C_3: & \ \ (x_3-8)^2+y_3^2&=&(2+x)^2&\ \ (b)\\ \text{square of length} \ C_0C_3: & \ \ (x_3-5)^2+y_3^2&=&(5-x)^2&\ \ (c) \\ \end{cases}$$

Expanding and subtracting (a) from (b) and (b) from (c) we obtain a linear system with $x$ and $x_3$ as unknowns, giving

$$x=30/19 \approx 1.579$$

together with $x_3=120/19$ and, after plugging these results into (a): $y_3=60/19$.

enter image description here


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