Circular Double Twins

Question

There are two identical blue circles and two identical orange circles arranged symmetrically in a larger red circle, as shown. The circles are tangent to each other where they touch.

If the radius of the red circle is then what is the radius of one of the orange circles?

Hint: Try finding a right triangle and applying the Pythagorean theorem.

Answer 1

Let the radius of the red circle be $R_1 $ and the radius of the orange circle be $R_2$

Then we can find in the figure

$O_3O_1=\frac{R_1}{2}+R_2$ $O_3C=R_1-R_2$ $O_1C=\frac{R_1}{2} $ and we also see that $O_3C$ is tangent of both the blue circles. So $\angle O_1CO_3=90$.

Therefore we can apply Pythagorean theorum for the right $\Delta O_1CO_3$

$\Rightarrow (R_1-R_2)^2+(\frac{R_1}{2})^2=(\frac{R_1}{2}+R_2)^2$

$\Rightarrow R_1^2+R_2^2-2R_1R_2+\frac{R_1^2}{4}=\frac{R_1^2}{4}+R_2^2+R_1R_2$

$\Rightarrow R_1^2=3R_1R_2$

$\Rightarrow R_2=\boxed{\frac{R_1}{3}}$