-GHEM is a rectangle because the rays are perpendicular to the tangents to the circle at the point of tangency, and the measure of angle E is 90 degrees (from the original rectangle). The fourth angle is a right angle, differentiating up to 360 degrees.[br]-Tangents drawn from an exterior point to a circle are congruent: EH=EM

- AB=x+y[br]- Tangents drawn from an exterior point to a circle are congruent: [br]EH=EM=x [br]DH=DK=DB=y[br]FM=FK=y–x[br]Result FD=y–x+y=2y–x

Pythagoras Theorem in triangle BDF: [math]\left(2y-x\right)^2=\left(x+y\right)^{_2}+y^2\Rightarrow4y^2-4xy+x^2=x^2+2xy+y^2[/math][br][math]2y^2=6xy,y\ne0\Rightarrow x=y=3x[/math]

[math]FB=4x;FK=2x\Rightarrow[/math][math]\Delta FBD\sim\Delta KFA\Rightarrow\frac{FB}{KF}=\frac{BD}{AK}\Rightarrow\frac{4x}{2x}=\frac{3x}{AK}\Rightarrow AK=\frac{3x}{2}\Rightarrow Area_{orange}=\frac{1}{2}\cdot\frac{9x^2}{4}\pi=\frac{9x^2}{8}[/math][br][math]Area_{blue}=x^2\pi[/math][br] [math]\frac{Area_{blue}}{Area_{orange}}=\frac{x^2\pi}{\frac{9x^2}{8}}=\frac{8}{9}[/math]