Is there a formula describing the length of line subtended from a chord in a circle? (Picture Included) Is there a formula that describes the length of line x in a circle, assuming you know the chord length (c), perpendicular distance of the chord from the circumference (d), circle radius (r), and the angle that line x subtends with the chord (a).





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*Observe that $MO = r -d$ and since the triangle $AMO$ is right angled, $$AM =\frac{c}{2}= \sqrt{OA^2 - OM^2} = \sqrt{r^2 - (r-d)^2}$$

*Triangle $MNO$ is right angled and $\angle \, OMN = a$ so 
$$MN = MO \,\cos(a) = (r-d) \,\cos(a), \,\,\,\, ON = MO \,\sin(a) = (r-d)\, \sin(a)$$ 

*Triangle $OND$ is right angled so by Pythagoras' theorem
$$ND = \sqrt{OD^2 - ON^2} = \sqrt{r^2 -(r-d)^2 \sin^2(a)}$$

*By the power of point relation
\begin{align}
 \frac{c^2}{4} &= AM^2 = AM \cdot MB = CM \cdot MD = CM \cdot \Big(MN + ND\Big)\\
&= x \, \Big( (r-d) \,\cos(a) + \sqrt{r^2 -(r-d)^2 \sin^2(a)}\Big)
\end{align} thus
$$x = \frac{c^2}{4 \, \Big( (r-d) \,\cos(a) + \sqrt{r^2 -(r-d)^2 \sin^2(a)}\Big)}$$ or equivalently, if you prefer
$$x = \frac{r^2 - (r-d)^2}{(r-d) \,\cos(a) + \sqrt{r^2 -(r-d)^2 \sin^2(a)}}$$
