Z9 – I – 5 MO 2018

Peter and Ivan created decorations from mutually identical white circles. Peter used four circles, which he placed so that each touched two other circles. Between them he then inserted another circle, which touched all four white circles, and he coloured it red.

Ivan used three circles, which he placed so that they touched each other mutually. Between them he then inserted another circle, which touched all three white circles, and he coloured it green. Ivan noticed that his green circle and Peter's red circle are of different sizes, and so they together began to find out how they differ.

Express the radii of the red and the green circle generally using the radius of the white circles.

Final Answer:

r₁ = (sqrt(2)-1) * r
r₂ = r( fr(2,3) * sqrt(3) - 1)

Step-by-step explanation:

$$\begin{gathered} (2r{)}^2+(2r{)}^2=(r+2r_1+r{)}^2 \\ 4r^2+4r^2=(2r+2r_1{)}^2 \\ 8r^2=(2r+2r_1{)}^2 \\ r\sqrt{8}=2r+2r_1 \\ {r}_1=(\sqrt{2}-1)\cdot r \end{gathered}$$

$$\begin{gathered} a=2r \\ h=\sqrt{3}/2\cdot a=\sqrt{3}\cdot r \\ 2/3h=r+r_2 \\ {r}_2=2/3\sqrt{3}\cdot r-r \\ {r}_2=r(\frac{2}{3}\cdot \sqrt{3}-1) \end{gathered}$$

Try another example