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.
[Math] Find radius of a circle from intersecting chords
circlesgeometry
Related Solutions
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Let's approach the problem in this way. First find the minimum length of a chord through $P.$ Using Pythagorean Theorem (the minimum-lengthed chord is perpendicular to the radius at $P$), we find this length to be $2\sqrt{10^{2} - 6^{2}} = 2 \cdot 8 = 16.$ Since this is an integer, it counts.
Next find the maximum length of a chord through $P.$ This is simply the diameter, which we know to be $20.$ Since this too is an integer, it also counts.
Notice that we can construct chords of any length between $16$ and $20$ that passes through $P.$ And for each such chord we draw through $P,$ we can reflect it across the radius through $P$ to produce a chord with equal length. This means that there are $2$ chords each of length $17,$ $18,$ and $19.$
Our final answer is $1 + 1 + 2 \cdot 3 = \boxed{8}.$
Best Answer
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.