Adam and Eve were creating decorations from identical white circles. Adam used four circles, arranging them so that each touched two other circles. He then inserted another circle between them, which touched all four white circles, and colored it red. Eve used three circles, arranging them so that they touched each other. She then inserted another circle between them, which touched all three white circles, and colored it green.

Eve noticed that her green circle and Adam's red circle were of different sizes, and they began to investigate how they differed. Express the radii of the red and green circles in terms of the radius of the white circles.

(M. Krejčová and M. Dillingerová)
![](https://cdn.mathpix.com/cropped/2024_04_17_1ca83b28e9c956579120g-6.jpg?height=500&width=1112&top_left_y=704&top_left_x=474)

Hint. What is the relationship between the centers of two touching circles and the point of contact?
In the following, we will repeatedly use the fact that the common point of two touching circles lies on the line connecting their centers. The radius of the white circles will be denoted by $r$.

The centers of Adam's white circles form the vertices of a square, and the center of the red circle lies at the center of this square, i.e., at the intersection of its diagonals. The diameter of the red circle is equal to the difference between the diagonal of the mentioned square and two radii $r$.

![](https://cdn.mathpix.com/cropped/2024_04_17_1ca83b28e9c956579120g-6.jpg?height=497&width=506&top_left_y=1733&top_left_x=775)

The side of the auxiliary square is $2r$, and its diagonal, according to the Pythagorean theorem, has the length

$$
\sqrt{4 r^{2} + 4 r^{2}} = 2 \sqrt{2} r.
$$

The radius of the red circle is therefore equal to

$$
\frac{1}{2}(2 \sqrt{2} r - 2 r) = (\sqrt{2} - 1) r.
$$

The centers of Eve's white circles form the vertices of an equilateral triangle, and the center of the green circle lies at the center of this triangle, i.e., at the intersection of its altitudes, or medians. The radius of the green circle is equal to the difference between the distance from the center, i.e., the centroid of the mentioned triangle, to its vertex and the radius $r$.

![](https://cdn.mathpix.com/cropped/2024_04_17_1ca83b28e9c956579120g-7.jpg?height=497&width=483&top_left_y=691&top_left_x=792)

The auxiliary equilateral triangle has a side length of $2r$, and each altitude divides it into right triangles with the hypotenuse $2r$ and one leg $r$. The other leg, i.e., the altitude, has the length, according to the Pythagorean theorem,

$$
\sqrt{4 r^{2} - r^{2}} = \sqrt{3} r.
$$

The distance from the centroid of an equilateral triangle to its vertex is equal to two-thirds of the length of the median, i.e., the altitude just expressed. The radius of the green circle is therefore equal to

$$
\frac{2}{3} \sqrt{3} r - r = \frac{2 \sqrt{3} - 3}{3} r.
$$