Compute the circumcenter (intersection of perpendicular bisectors), circumradius, and triangle classification. Visualize the circumscribed circle, vertices, and centers on an interactive canvas.
The circumcenter (denoted O) is the unique point equidistant from all three vertices of a triangle. It is the center of the circumcircle — the circle that passes through A, B, and C. The circumcenter lies at the intersection of the three perpendicular bisectors of the sides. This classical result appears in Euclid's Elements and is fundamental to triangle geometry, trigonometry, and computational geometry.
For vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃):
|OA| = |OB| = |OC| = R (circumradius)
Perpendicular bisector of AB: (x₂−x₁)(x − xₘ) + (y₂−y₁)(y − yₘ) = 0
The circumcenter's properties were explored by ancient Greek geometers, but the systematic study of triangle centers blossomed in the 18th century with Leonhard Euler, who discovered that the circumcenter O, centroid G, and orthocenter H are always collinear (Euler line) with the relation OG : GH = 1 : 2. Moreover, the circumcenter is the isogonal conjugate of the orthocenter. In an acute triangle, O lies inside; in a right triangle, O coincides with the midpoint of the hypotenuse; in an obtuse triangle, O lies outside.
The circumradius R connects deeply with the law of sines: a/sin A = b/sin B = c/sin C = 2R, where a, b, c are side lengths. This law is indispensable in surveying, astronomy, and navigation.
Our calculator follows an analytic geometry approach:
All calculations are performed with double precision, ensuring accuracy up to 10⁻¹² relative error.
Global Navigation Satellite Systems (GNSS) use the principle of circumcenters: given three reference points (satellites) and distances, the intersection of perpendicular bisectors of the chord segments defines the receiver's position. Similarly, in computer graphics, the circumcenter is vital for constructing Delaunay triangulations, which maximize the minimum angle and avoid sliver triangles. Our calculator mimics the underlying geometry used in such high‑precision algorithms.
Example: Points A(0,0), B(10,0), C(5,8) produce circumcenter at (5.00, 3.75) and R ≈ 5.15. This unique center balances all three vertices, enabling optimal circle packing and interpolation.
| Triangle Type | Example Vertices | Circumcenter O | Circumradius R | Location of O |
|---|---|---|---|---|
| Acute Scalene | (0,0), (5,0), (2,4) | (2.50, 1.81) | 2.90 | Inside triangle |
| Right (∠C=90°) | (0,0), (4,0), (0,3) | (2.00, 1.50) | 2.50 | Midpoint of hypotenuse |
| Obtuse | (0,0), (6,0), (2,1) | (3.00, -4.00) | 5.00 | Outside triangle |
| Equilateral | (0,0), (2,0), (1,1.732) | (1.00, 0.577) | 1.155 | Same as centroid |