Circumcenter and Circumcircle
The circumcenter of a triangle is the unique point where the three perpendicular bisectors
of the sides intersect. A perpendicular bisector is a line that passes through the midpoint of a segment and is perpendicular
to that segment. A key property is that every point on a perpendicular bisector is equidistant from the segment’s endpoints.
Because the circumcenter lies on the perpendicular bisector of \(AB\), it satisfies \(OA=OB\). Likewise, lying on the bisectors of
\(BC\) and \(CA\) forces \(OB=OC\) and \(OC=OA\). Together, these equalities mean the circumcenter \(O\) is the single point
that is the same distance from all three vertices \(A\), \(B\), and \(C\).
That common distance is called the circumradius \(R\). The circle centered at \(O\) with radius \(R\) passes
through all three vertices; it is the triangle’s circumcircle. In coordinate form, if \(O=(h,k)\), the circumcircle
equation is
\[
(x-h)^2 + (y-k)^2 = R^2.
\]
This tool computes \(O\) and \(R\), then writes the circle equation and displays the triangle, bisectors, and circumcircle in an
interactive plot.
The location of the circumcenter depends on the triangle type. In an acute triangle, the perpendicular bisectors
intersect inside the triangle, so the circumcenter is inside. In a right triangle, the circumcenter is exactly the
midpoint of the hypotenuse (a classic result), so it lies on the triangle’s boundary. In an obtuse triangle, the
circumcenter lies outside the triangle because at least one perpendicular bisector must intersect outside when an angle exceeds \(90^\circ\).
There are several useful formulas for the circumradius. If you know a side \(a\) and the opposite angle \(A\), then
\[
R=\frac{a}{2\sin A}.
\]
If you know all three side lengths \(a,b,c\) and the area \(\Delta\), then
\[
R=\frac{abc}{4\Delta}.
\]
This second form pairs nicely with Heron’s formula, which computes the area from side lengths using the semiperimeter
\(s=\frac{a+b+c}{2}\) and \(\Delta=\sqrt{s(s-a)(s-b)(s-c)}\). In practice, these formulas are also a good way to cross-check results.
Finally, note the main existence condition: a circumcenter is well-defined only when the triangle is non-degenerate.
If the three points are collinear (or the side lengths violate the triangle inequality), the “triangle” has zero area and there is no
unique circumcircle. This calculator detects such cases and reports a clear error instead of producing misleading numbers.
