A circle with diameter $AC$ is intersected by a secant at points $B,D$. The secant and the diameter intersect at point $P$ outside the circle. 
A circle with diameter $AC$ is intersected by a secant at points $B,D$. The secant and the diameter intersect at point $P$ outside the circle. Perpendiculars $AE$ and $CF$ are drawn at the secant. If $EB = 2$ and $BD = 6$, find $DF$.

What I Tried: Here is a picture :-

Unfortunately I have no idea to start with this. A first idea is to find similar triangles, which I find a lot, but I cannot find a way on how to use the lengths of $EB$ and $BD$.
Can anyone help me? Thank You.

$\angle ADE = \angle DCF\Rightarrow \triangle AED \sim \triangle DFC $
$$\Rightarrow \dfrac{AE}{DF} = \dfrac{AD}{DC} $$
$\angle ABE = \angle ACD\Rightarrow\triangle AEB \sim \triangle ADC $
$$\Rightarrow \dfrac{AE}{EB} = \dfrac{AD}{DC} $$
$$\therefore DF=EB =2$$
