Circumcenter Calculator
Calculate the circumcenter and circumradius of a triangle from vertex coordinates
Triangle Vertices
First vertex (x₁, y₁)
Second vertex (x₂, y₂)
Third vertex (x₃, y₃)
Circumcenter Results
Step-by-Step Solution
Example Calculation
Right Triangle Example
Problem: Find the circumcenter of triangle with vertices at A(0,0), B(3,0), and C(0,4).
Note: This is a right triangle with the right angle at vertex A.
Solution
Given: A(0,0), B(3,0), C(0,4)
Calculate J: (0-3)(0-4) - (0-0)(0-0) = 12
Calculate t: 0² + 0² - 3² - 0² = -9
Calculate u: 0² + 0² - 0² - 4² = -16
Circumcenter X: -((0-0)(-16) + (0-4)(-9)) / (2×12) = 1.5
Circumcenter Y: ((0-3)(-16) - (0-0)(-9)) / (2×12) = 2
Result: Circumcenter at (1.5, 2), which is the midpoint of hypotenuse BC.
Circumcenter Formula
Given vertices:
Calculate:
Circumcenter:
Circumcenter Location
Understanding the Circumcenter
What is a Circumcenter?
The circumcenter of a triangle is the center of the circumscribed circle (circumcircle) - the unique circle that passes through all three vertices of the triangle. It's the point where the perpendicular bisectors of the triangle's sides intersect.
Key Properties
- •Equidistant from all three vertices
- •Center of the triangle's circumcircle
- •Location depends on triangle type
- •Every triangle has exactly one circumcenter
Mathematical Derivation
The circumcenter formula is derived from the condition that it must be equidistant from all three vertices. Using the distance formula and solving the system of equations D₁ = D₂ = D₃ leads to the coordinate formulas.
Applications
- •Geometric constructions and proofs
- •Computer graphics and CAD systems
- •Navigation and GPS triangulation
- •Engineering and architectural design
Fun Fact: The circumcenter, centroid, and orthocenter of any triangle are collinear and lie on the Euler line!