This question was uploaded on 03/01/22 on social media accounts.
Solution 1:
Green length:
\[(r+a)(r-a)=x^2\]
\[\Rightarrow r^2-a^2=x^2\]
Intersecting chord theorem in red circle:
\[(2r+2\sqrt2+a)(2r-2\sqrt2-a)=x^2\]
\[\Rightarrow 2r^2-a^2-2ar\sqrt2=x^2\]
From both equations:
\[r^2-a^2=2r^2-a^2-2ar\sqrt2\]
\[\Rightarrow a=\frac{r}{2\sqrt2}\]
\[\Rightarrow x=r\sqrt\frac72\]
\[\Rightarrow Green=2x=2r\sqrt\frac72\]
Orange length:
Triangle with sides (k, y, 2r) is right angled triangle
\[\Rightarrow k^2+y^2=(2r)^2\]
\[\Rightarrow (\sqrt2-1)^2+y^2=(2r)^2\]
\[\Rightarrow y=\sqrt{1+2\sqrt2}\]
\[\Rightarrow Orange=2y=2\sqrt{1+2\sqrt2}\]
Ratio:
\[\frac{Green}{Orange}=\frac{\sqrt{7/2}}{2\sqrt{1+2\sqrt2}}\]
Alternate forms:
\[Ratio=\frac12\sqrt\frac{7}{2+4\sqrt2}\]
\[Ratio=\frac12\sqrt{\sqrt2-\frac12}\]
\[Ratio=\sqrt{\sqrt\frac18-\frac18}\]
