I'm looking for way to find the length of a secant line intersecting another line through the center of a circle with a known radius. The intersection point is on the circle and the angle between 2 lines is also known. Attached image should clarify:
p is the intersection point of the two lines pd and pe on the circle.
pe is the line through the center and is 2 x radius (known).
a is the angle (known).
I would like to know an easy way to find the length of pd, and ideally also the length of pf intersecting with the tangent of circle parallel to pe.
I found the intersecting secants theorem, but this is of no use as the intersection point is outside the circle.
The triangle PED having one side on the diameter of the circle is a right triangle.
If the angle of DPE is $\alpha$ some trigonometry tolds you:
$$\cos{\alpha}=\frac{PD}{2r}$$ $$PD=2r\cos{\alpha}$$
For $PF$ as André Nicolas stated you can see this:
The lenght of FG is equal to the radius (because the line tangent is parallel to the diameter, therefore some trigonometry again gives:
$$\sin{\alpha}=\frac{r}{PF}$$ $$PF=\frac{r}{\sin{\alpha}}$$