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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.

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1 Answer

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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:

$$\sin{\alpha}=\frac{r}{PF}$$ $$PF=\frac{r}{\sin{\alpha}}$$

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