Secant line and diameter of a circle A secant line incident to a circle at points $A$ and $C$ intersects the circle's diameter at point $B$ with a $45^\circ$ angle. If the length of $AB$ is $1$ and the length of $BC$ is $7$, then what is the circle's radius?


We can set up a system of equations for $BE$ and $BD$, then that will give us the radius, which is 
$$r=\frac{BE+BD}{2}$$
First, by the chord-chord theorem (AKA the power chord theorem), we can write
$$AB\times BC=BE\times BD$$
$$\Rightarrow BE\times BD=7\quad\quad\quad(1)$$
Next, we can use the cosine law in $\triangle BCO$ (note that $CO$ is also a radius):
$$CO^2=BC^2+BO^2-2\times BC\times BO\times \cos \angle CBO$$
$$\Rightarrow r^2=49+(BE-r)^2-14(BE-r)\cos(45)$$
$$\Rightarrow r^2=49+(BE-r)^2-7\sqrt{2}(BE-r)$$
$$\Rightarrow \left(\frac{BE+BD}{2}\right)^2=49+\left(\frac{BE-BD}{2}\right)^2-7\sqrt2\left(\frac{BE-BD}{2}\right)$$
$$\dots$$
$$\Longrightarrow BE\times BD+\frac{7\sqrt2 BE}{2}-\frac{7\sqrt2 BD}{2}=49$$
$$\Rightarrow \left(BE-\frac{7\sqrt2}{2}\right)\left(BD+\frac{7\sqrt2}{2}\right)=\frac{49}{2}\quad\quad\quad(2)$$
Can you continue the algebra?
