# IMC 2011 question involving six tangent circles

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.

• Please draw a sketch showing the circles contacting externally.. – Narasimham Mar 26 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 $$r=\frac{3+\sqrt{17}}{2}.$$

• Yes, thanks a lot mate, you're a genius – kenith Mar 3 at 12:18