I have a red circle with a radius of 1. I'd like to get the exact ratio between it and the bigger blue circle radius. So how do you calculate AI/AJ in this case? I suppose it going to involve square root 3 anyway, but I can't get an idea, how to "mathematize" this diagram.
We have $$ \begin{align} \overrightarrow{AH}&=\begin{pmatrix}1\\0\end{pmatrix}+2\cdot\begin{pmatrix}1/2\\\sqrt3/2\end{pmatrix}\\ &=\begin{pmatrix}2\\\sqrt{3}\end{pmatrix} \end{align} $$ and since $|\overrightarrow{AH}|=\sqrt 7$ we get $$ \begin{align} \overrightarrow{AJ}&=\overrightarrow{AH}+\frac{\overrightarrow{AH}}{|\overrightarrow{AH}|}\\ &=\begin{pmatrix}2+2/\sqrt 7\\\sqrt{3}+\sqrt{3}/\sqrt{7}\end{pmatrix} \end{align} $$ which has squared length $$ \begin{align} |\overrightarrow{AJ}|^2&=4+4/7+8/\sqrt{7}+3+3/7+6/\sqrt 7\\ &=8+14/\sqrt{7} \end{align} $$ And $14/\sqrt 7=14\sqrt7/7=2\sqrt 7$. Then just take the square root of that to get $$ |\overrightarrow{AJ}|=\sqrt{8+2\sqrt 7}\approx 3.6458 $$
The ratio you were asking for is then $$ \begin{align} \frac{|\overrightarrow{AI}|}{|\overrightarrow{AJ}|}&=\frac{1}{\sqrt{8+2\sqrt 7}}\\ &=\frac{\sqrt{8+2\sqrt 7}}{8+2\sqrt 7}\\ &=\frac{\sqrt{8+2\sqrt 7}(8-2\sqrt 7)}{36}\\ &=\frac{\sqrt{8+2\sqrt 7}(4-\sqrt 7)}{18}\\ &\approx 0.274291885177431765 \end{align} $$