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Say I have two chords that intersect inside a circle, not at a right angle, and neither is the diameter. It seems to me this is enough information that the circle must be unique, but I can't seem to find the radius.

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The center of the circle is at the intersection of the perpendicular bisectors of the two chords. Any of the segments joining this point to an endpoint of a chord will be a radius.

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Let $x$ be the known distance between circle center and point of chords intersection.

You must search for the useful / relevant property.

$$OB \cdot OD = OA\cdot OC = P$$

By property of intersecting chords ( segments product P= a known constant)..

$$ (R+x)(R-x)=P $$

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from which radius of circle $R$ can be found

$$ R= \sqrt{x^2 + P}. $$

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