How this calculator works
The perpendicular bisectors of two sides meet at the circumcentre. Translating the first vertex to the origin reduces cancellation when the whole triangle is far from the coordinate origin. Solving the two equal-distance equations gives the centre offset. The distance table compares all three radii. The squared-side test classifies the triangle and therefore locates its centre; relative differences within 10^-12 are treated as right.
Formula
With B-A=(u,w), C-A=(p,q), D=2(uq-wp): ox=[(u²+w²)q-(p²+q²)w]/D; oy=[u(p²+q²)-p(u²+w²)]/D; O=A+(ox,oy).
Worked example
For A=(0,0), B=(6,0), C=(0,4), the centre is (3,2), the midpoint of the hypotenuse. Each vertex is √13 units from it, and the triangle area is 12 square units.
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Frequently asked questions
Why can the circumcentre be outside?
For an obtuse triangle the perpendicular bisectors meet outside the outline. A circumcircle still passes through every vertex.
What happens for three points on a line?
There is no finite circle through three distinct collinear points. The determinant is zero and the calculator rejects the coordinates.
How is this different from the circumscribed circle tool?
That tool starts with side lengths. This one uses vertex coordinates to locate the centre in the coordinate plane and compare its three vertex distances.