Circumcenter calculator

When the three points lie on a straight line there is no finite circumcircle, so the tool returns invalid_input instead of Infinity or a silent NaN. Through the API a successful request costs $0.002; validation failures are not charged.

What the circumcenter is and what the response contains

The preferred inputs are the six scalars ax, ay, bx, by, cx, and cy as JSON numbers; decimal strings are accepted for form and spreadsheet callers, and nested points a, b, c as [x, y] or {x, y} are also read. The response echoes the three vertices, returns circumcenter as an {x, y} object, circumradius and diameter as numbers, the triangle area K, and the side lengths ab, bc, and ca. A collinear boolean is always false on success because collinear input is rejected before a circle is formed. Missing coordinates, non-finite values, coincident vertices, and collinear triples all raise invalid_input with a message that names the failing condition. That discipline keeps automated tests stable, makes caching safe, and prevents a student or a script from treating a degenerate triple as a circle of infinite radius.

How coordinates are read and how the closed form is evaluated

You may also pass points or vertices as an array of three [x, y] pairs. After parsing, pairwise distances are checked so that coincident vertices fail fast with a message naming the pair. Otherwise the circumcenter coordinates follow the standard expansion: Ox equals the weighted sum of squared norms times the opposite y-differences divided by D, and Oy equals the analogous sum with x-differences. Circumradius is the distance from O to A; an independent check R equals abc over four K uses the product of side lengths and the triangle area K equals absolute D over four, and the two estimates are averaged for a stable reported radius under floating-point noise. The engine resolves vertex A from ax and ay (aliases x1, a_x, y1, a_y) or from nested field a; likewise B from bx and by and C from cx and cy. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the bisector algebra yourself. Near-degenerate triangles with very small area can amplify floating-point error; if consistency between vertex distances fails, the engine surfaces invalid_input rather than a silently inconsistent circle.

Where circumcenter solving fits in study, design, and code

On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split—free interactive exploration and a priced, identical API—matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with unit converters when coordinates arrive in mixed units, with other free geometry endpoints when you need SSA or side-angle companions for a lab of deterministic tools, and with documentation that shows both the bisectors and the finished circle so the numbers and the picture stay aligned for learners and reviewers.