Input triangle vertex coordinates to calculate all five triangle centers
Vertex A
x₁
y₁
Vertex B
x₂
y₂
Vertex C
x₃
y₃
Calculation Results
Detailed Step-by-Step Derivation
Triangle Center Formula Summary
Centroid G:G((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
Orthocenter H:Intersection of three altitudes solved by linear equations
Circumcenter O:Equidistant from three vertices, solved by distance equation system
Incenter I:Equidistant from three sides, derived from angle bisector properties
The five triangle centers are fundamental geometric feature points with unique definitions and calculation methods. All five centers coincide perfectly in an equilateral triangle.
⚠Note: Three vertices cannot be collinear, otherwise a valid triangle cannot be formed. All centers overlap in equilateral triangles; isosceles triangles share symmetry for certain centers.
Understanding the Five Triangle Centers
Centroid, orthocenter, circumcenter, incenter and excenter are core concept points in plane geometry, each carrying distinct geometric definitions and mathematical properties.
Centroid
Intersection of three medians, dividing each median into a 2:1 ratio. It serves as the triangle’s mass balance point with the simplest coordinate formula among all five centers.
Orthocenter
Intersection of three altitude lines. It lies inside acute triangles, outside obtuse triangles, and exactly at the right-angle vertex of right triangles.
Circumcenter
Intersection of three perpendicular bisectors and center of the circumscribed circle. Inside acute triangles, outside obtuse triangles, and at the hypotenuse midpoint of right triangles.
Incenter
Intersection of three internal angle bisectors and center of the inscribed circle. Always located inside the triangle and equally distant from all three sides.
💡 Equilateral Triangle Special Case: All five triangle centers merge into one single point. For vertices (0,0), (a,0), (a/2, a√3/2), all centers locate at (a/2, √3a/6).
Centroid is the intersection of three medians with average coordinate formula. Orthocenter comes from three altitude intersections. Circumcenter solves equal distance to three vertices. Incenter uses equal distance to three sides. Excenter forms from one internal and two external angle bisectors.
What key properties does the centroid have?▼
The centroid divides every median into a fixed 2:1 ratio from vertex to midpoint. It acts as the geometric balance center and has the simplest average coordinate formula among all triangle centers.
What is the difference between incenter and circumcenter?▼
The incenter is the inscribed circle center equidistant from all three sides. The circumcenter is the circumscribed circle center equidistant from all three vertices. The circumcenter lies outside obtuse triangles while the incenter always stays inside.
Free online calculators and tools covering mathematics, unit conversion, text processing, and daily life. Accurate, fast, mobile-friendly, and completely free to use.