Two circles $A$ and $B$, both with radius $1$, touch each other externally. Four circles $P$, $Q$, $R$, and $S$, all with the same radius $r$, are such that $P$ touches $A$, $B$, $Q$, $S$ externally; $Q$ touches $P$, $B$, and $R$ externally; $R$ touches $A$, $B$, $Q$, and $S$ externally; and $S$ touches $P$, $A$, and $R$ externally. Calculate $r$.

Context: The question above is from the 2011 IMC (International Mathematics Competition) exam. The solutions do not exist anywhere on the internet. Personally, I believe that the solution to this question will help the geometrical thought of the users of this website.

Attempt: I tried solving the question above by connecting the radii; however, my efforts did not prosper.

Can you guys please help me?

  • $\begingroup$ Please draw a sketch showing the circles contacting externally.. $\endgroup$ – Narasimham Mar 26 '19 at 6:40

Let $A$, $B$, $S$, $P$, $Q$ and $R$ be centers of our circles.

Thus, by the symmetry $\{A,B\}\subset SQ.$

Also, let $PM$ be an altitude of the $\Delta SPQ$ and $\Delta APB.$

Thus, $$PM^2=SP^2-SM^2=AP^2-AM^2,$$ which gives $$(2r)^2-(2+r)^2=(1+r)^2-1^2.$$ Can you end it now?

I got $r=\frac{3+\sqrt{17}}{2}.$

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  • $\begingroup$ Yes, thanks a lot mate, you're a genius $\endgroup$ – Michalis Christofi Mar 3 '19 at 12:18

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