Colourful Square

Geometry Level 5

Let \(\color{red}{R},\color{blue}{B_L},\color{orange}{Y},\color{green}{G},\color{brown}{B_R}\) denote the areas of red circle, blue circle, yellow, green, brown regions respectively.If the side of the square is \(\sqrt{2}+1\) units, evaluate the value of the following expression to correct four decimal places:

\[\dfrac{\color{blue}{B_L}(\color{brown}{B_R}+\color{blue}{B_L})}{\color{red}{R}\color{orange}{Y}} +4\color{green}{G}\]

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