Find the circumcenter, circumradius, centroid, incenter, orthocenter, perpendicular bisector equations, area, and perimeter of a triangle from 3 vertex coordinates.
The <strong>circumcenter</strong> of a triangle is the point where all three perpendicular bisectors of the sides meet. It is equidistant from all three vertices, making it the center of the <em>circumscribed circle</em> (circumcircle) — the unique circle passing through all three vertices. Finding the circumcenter is essential in coordinate geometry, triangulation algorithms, mesh generation, and surveying.
This calculator takes six inputs — the (x, y) coordinates of vertices A, B, and C — and instantly computes the circumcenter, circumradius, centroid, incenter with inradius, and orthocenter. It also provides perpendicular bisector equations, the Euler-line distance, all three side lengths and angles, the triangle area (Heron's formula), and the perimeter.
A color-coded comparison chart shows all four classic triangle centers side by side, and a detailed table explains which centers always lie inside the triangle and which can move outside for obtuse triangles. Eight preset buttons let you explore common triangle types (right, equilateral, isosceles, obtuse, scalene) with a single click. An advanced section reveals side lengths, angles, semi-perimeter, and perpendicular bisector equations.
Whether you're solving a coordinate-geometry homework problem, studying for a competition, or implementing a computational geometry algorithm, this tool gives you every triangle-center measurement in one place.
This circumcenter of a triangle calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Vertex A — x₁, Vertex A — y₁, Vertex B — x₂, Vertex B — y₂ and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
Circumcenter: Ux = [(x₁²+y₁²)(y₂−y₃) + (x₂²+y₂²)(y₃−y₁) + (x₃²+y₃²)(y₁−y₂)] / D, where D = 2[x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)]. Circumradius R = distance(U, any vertex). Centroid G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). Incenter I = (a·x₁+b·x₂+c·x₃)/(a+b+c). Orthocenter H = A+B+C − 2·U.
Result: For x1=0, y1=0, x2=6, the tool returns the solved circumcenter of a triangle outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in circumcenter of a triangle formulas and reports derived values, checks, and classifications automatically.
This page is tailored to circumcenter of a triangle, with outputs tied directly to the form fields (Vertex A — x₁, Vertex A — y₁, Vertex B — x₂, Vertex B — y₂). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
The circumcenter is the point equidistant from all three vertices. It is the center of the circumscribed circle that passes through all three vertices.
Construct the perpendicular bisector of any two sides. Their intersection point is the circumcenter.
No. It is inside for acute triangles, on the hypotenuse for right triangles, and outside for obtuse triangles.
The circumcenter is equidistant from the three vertices (center of circumscribed circle). The incenter is equidistant from the three sides (center of inscribed circle).
The Euler line passes through the circumcenter, centroid, and orthocenter. The centroid divides the segment from circumcenter to orthocenter in a 1:2 ratio.
Yes. The formulas work for any real-valued coordinates. Try the "Negative coords" preset to see an example.