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Illustration of the problem where R and S are the length and width respectively. A is 4 cm and B is 9 cm.

I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius $4 \, cm$ and touches one side of the box, the middle sized one has radius $9 \, cm$ and touches two sides of the box and the largest one touches three sides of the box. What is the radius of the largest one?

In this case circle $A$ has radius $4 \, cm$ and circle $B$ has radius $9 \,cm$.

I have tried to look at the problem from different points of view as well as by investigating for a pattern, however the answer is not visibly yielded this way. I do not know what implications the circles touching each other has on the answer, but this must be a key observation to understand. Otherwise, I have tried to link the circle centres together forming a triangle, however this seems to lead nowhere. However I may be wrong.

Thank you for the help.

Rob Arthan
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Tom Finet
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1 Answers1

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Let the unknown radius be $r$. Then the vertical distance between the centres of $B,C$ is $r-9$, hence by Pythagoras their horizontal distance is $\sqrt{(9+r)^2-(r-9)^2}=6\sqrt{r}$. Likewise, the vertical distance between centres of $A,C$ is $r-4$ and therefore the horizontal distance is $\sqrt{(4+r)^2-(r-4)^2}=4\sqrt r$. Thus the horizontal distance between centres of $A,B$ is $6\sqrt r-4\sqrt r=2\sqrt r$ and the vertical distance is $\sqrt{13^2-4r}$, but it is also $2r-13$. We conclude $$13^2-4r=(2r-13)^2 $$ and so ($r=0$ or) $$r=12. $$

Hagen von Eitzen
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