Euler Line Calculator
Circumcenter, Centroid, Orthocenter alignment
Input Method
Euler's Line Theorem
The circumcenter (O), centroid (G), and orthocenter (H) are collinear. G divides OH in ratio 1:2.
Euler Line Visualization
Understanding the Euler Line
The Euler Line is one of the most beautiful results in triangle geometry, discovered by Leonhard Euler in 1765. It states that three important triangle centers are always collinear (lie on the same line).
The Four Centers on the Euler Line
Circumcenter (O)
Center of the circumscribed circle. Equidistant from all vertices.
Centroid (G)
Intersection of medians. Center of mass of the triangle.
Orthocenter (H)
Intersection of altitudes. Position varies by triangle type.
Nine-Point Center (N)
Center of the nine-point circle. Midpoint of OH.
Key Relationships
- OG : GH = 1 : 2 - The centroid divides OH in ratio 1:2
- H = 3G - 2O - Vector relationship between centers
- N = (O + H) / 2 - Nine-point center is midpoint of O and H
Special Cases
- Equilateral triangle: All four centers coincide at a single point
- Isosceles triangle: Euler line is the axis of symmetry
- Right triangle: Orthocenter is at the right angle vertex
Frequently Asked Questions
Who discovered the Euler line?
Leonhard Euler discovered this remarkable property in 1765, along with the 1:2 ratio of the centroid's position.
Is the incenter on the Euler line?
No! The incenter (center of inscribed circle) is generally NOT on the Euler line, except for isosceles triangles.
What is the nine-point circle?
The nine-point circle passes through 9 special points: the 3 midpoints of sides, 3 feet of altitudes, and 3 midpoints of segments from vertices to orthocenter.
Why is the ratio 1:2 significant?
The 1:2 ratio (G divides OH) is connected to how the centroid divides medians in ratio 2:1 from vertex to midpoint.