Circle through three points

When the three points lie on a straight line there is no finite circle through all three, 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 a circle through three points is and what the response contains

In the Euclidean plane, any three points that do not lie on a single straight line determine exactly one circle. That circle is the set of all points at a fixed distance R from a unique center C; the three given points are chords of the circle, so C is equidistant from all three and must lie at the intersection of the perpendicular bisectors of any two sides of the triangle they form. This tool exists to turn raw coordinates into that center and radius without forcing you to set up bisector equations by hand or to expand the general circle equation x squared plus y squared plus D x plus E y plus F equals zero and solve a three-by-three system yourself. If you need a least-squares fit through many noisy samples, a circle in three dimensions, or a different triangle center such as the incenter, use a tool dedicated to those problems; the search intent here is the exact circle through three coordinates and a clear answer of center, radius, or an honest error. The preferred inputs are the six scalars x1, y1, x2, y2, x3, and y3 as JSON numbers; decimal strings are accepted for form and spreadsheet callers, and nested points p1, p2, p3 as [x, y] or {x, y} are also read, as is a points array of three pairs. The response echoes the three points, returns center as an {x, y} object, radius and diameter as numbers, radius_squared for equation work, a collinear flag that is always false on success, a human-readable equation string of the form (x − h) squared plus (y − k) squared equals R squared, and a formula field that records the determinant identity used. Missing coordinates, non-finite values, coincident points, 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

The engine resolves the first point from x1 and y1 (aliases ax, a_x, p1x and ay, a_y, p1y) or from nested field p1 or a; likewise the second point from x2 and y2 and the third from x3 and y3. 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 triples with very small area can amplify floating-point error; if consistency between point-to-center distances fails, the engine surfaces invalid_input rather than a silently inconsistent circle.

Where finding a circle through three points fits in study, design, and code

Students meet this construction when a textbook asks for the circle through three given points, when a geometry exam demands the center equidistant from three map pins, or when analytic geometry homework wants the general circle equation from three coordinate pairs. Teachers can pre-generate answer keys that list center and radius for integer-coordinate triples and refuse collinear inputs instead of printing Infinity. Game and simulation teams use the three-point circle as a cheap bounding circle for a triangle mesh face, for hit-test regions, or for placing a light or camera that must sit equally far from three landmarks. Developers call the same contract from scripts to seed deterministic test vectors for geometry libraries, to illustrate perpendicular bisectors in teaching tools, or to satisfy programming-challenge statements that demand the circle through three coordinates. 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 line intersections or triangle side-angle companions for a lab of deterministic tools, and with documentation that shows both the chords and the finished circle so the numbers and the picture stay aligned for learners and reviewers. Whether you arrived searching for a circle through three points or for the center and radius of that construction, the contract is the same: six coordinates in, a unique circle or a clear invalid_input out.