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**SlipEternal** Let $r$ be the radius of the circle. Let $y$ be the length of the versine. Then the distance from the point where $x$ touches the circle to the center of the circle is $r$ (it is a radius). If we were to assign coordinates to that point (with the circle centered at the origin), then the point where the segment labeled $x$ intersects the segment labeled $y$ would have coordinates $(r-y,0)$. Then the point where the segment labeled $x$ intersects the circle would have coordinates $(r-y+x\cos \theta,x\sin \theta)$. This means that:

$$(r-y+x\cos \theta)^2+(x\sin \theta)^2 = r^2$$

Solving for $r$ gives:

$$r = \dfrac{x^2+y^2-2xy\cos \theta}{2y-2x\cos \theta}$$

So long as $y \neq x\cos \theta$. However, that only happens when $\theta=0$ and $x=y$.