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Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.


Age 16 to 18 Challenge Level:

This solution is from Yatir Halevi.

Let $a$, $b$ and $c$ be the radii the circles centered at $A$, $B$ and $C$ respectively.

Angle $BOC = 90^\circ$ since, $OB$ is tangent to the circle centered at $C$ and hence it is perpendicular to the radius. That also means that the radius of the the circle centered at $O$ is $2c$.

As $BC$ connects $B$ and $C$ it is equal to $b+c$.

By Pythagoras theorem: $(2c-b)^2 + c^2 = (b+c)^2$. Hence $4c^2=6bc$ and $c$ can't be equal to $0$ so $2c=3b$.

We could extend the lines $AC$ and $BO$, to create a mirror image, where we have two circles centered at $A'$ and $B'$ with radii $a'$ and $b'$ respectively, and due to symmetry issues, $a'=a$ and $b'=b$. We will also connect $O$ with $A'$ and $O$ with $A$.

Now $OA=OA'= 2c-a$ and $AC = A'C = c+a$ and so the line $OC$ cuts $A'A$ in half, and because the sides of the triangle $A'OA$ are equal, $OC$ is also perpendicular to $A'A$, hence $\angle OCA = 90$.

As triangle $ACO$ is a right-angled triangle, using Pythagoras' Theorem: $AC^2+OC^2=AO^2$ and so $$(a+c)^2 + c^2 =(2c-a)^2$$ which simplifies to $6ac=2c^2$ and $c$ can't be equal to $0$ so $c=3a$. Combining this with $2c=3b$ we get: $3b=6a$ that is $b=2a$ and so $a=b/2=c/3$, that is they are in the ratio $1:2:3$ as required.

If we mirror the image in the line $BO$ thus creating a new circle below the one centered at $B$, centered at $A''$ it has the same radius as $A$ by symmetry. By the the same line of thought that we used before, we reach the conclusion that $\angle ABO=90$. What we get is that $CABO$ has 3 right angles, meaning that the fourth is also a right angle and that it is a rectangle and that triangle $ABC$ is also right angled.

Let's look at triangle $OBC$ in which $OB = 2c-b = 6a - 2a = 4a$, $OC = c = 3a$ and $BC = b + c = 2a+3a = 5a.$ So the ratio between the sides are $3:4:5$, like we wanted to prove.

Let's look at triangle $ABC$ in which $AB = a + b = a + 2a = 3a$, $AC = a + c = a + 3a = 4a$ and $CB = b + c = 2a + 3a = 5a$. So the ratio between the sides are $3:4:5$, like we wanted to prove.

Arun Iyer proved this result by applying Descartes Circle Theorem.