Euler Line Calculator

Construct circumcenter, centroid, orthocenter and Euler-line direction from planar vertices; equilateral coincident centers leave no unique Euler line.

Description

The Euler line relates a triangle's circumcenter, centroid and orthocenter.

The Euler line relates a triangle's circumcenter, centroid and orthocenter. Coincident centers, as in an equilateral triangle, leave no unique line direction.^[1]

Construct circumcenter, centroid, orthocenter and Euler-line direction from planar vertices; equilateral coincident centers leave no unique Euler line.

When to use Euler Line Calculator

Use this tool to construct circumcenter, centroid, orthocenter and Euler-line direction from planar vertices; equilateral coincident centers leave no unique Euler line.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Euler Line Calculator and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter vertices using the field table and calculation rules, then run the tool.
Read circumcenter, centroid, orthocenter, eulerLineDirection, eulerLineDeterminate. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
vertices	Vertices	list of list of number	Required

Enter three well-conditioned noncollinear planar vertices. Compute all three centers and return the unnormalized direction vector. Mark the line determinate only when direction magnitude exceeds 10⁻¹² times circumradius.

direction = orthocenter − circumcenter

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
circumcenter	Circumcenter	list of number
centroid	Centroid	list of number
orthocenter	Orthocenter	list of number
eulerLineDirection	Euler line direction	list of number
eulerLineDeterminate	Euler line determinate	boolean

json

{
  "input": {
    "vertices": [
      [
        0,
        0
      ],
      [
        4,
        0
      ],
      [
        0,
        3
      ]
    ]
  },
  "output": {
    "circumcenter": [
      2,
      1.5
    ],
    "centroid": [
      1.3333333333333333,
      1
    ],
    "orthocenter": [
      0,
      0
    ],
    "eulerLineDirection": [
      -2,
      -1.5
    ],
    "eulerLineDeterminate": true
  }
}

Limitations and assumptions

The direction is not a unit vector. Nearly collinear triangles are rejected, and coincident centers return a false determinate flag.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Matrices are limited to 128 rows and 128 entries per row; other numeric lists are limited to 100,000 entries. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use Nine Point Circle Calculator: Construct nine-point circle and actual triangle side midpoints; center uses circumcenter/orthocenter and radius is half circumradius, not a generic circle from radius. For a complementary task, use Napoleon Triangle Area Calculator: Compute outer/inner Napoleon triangle areas from actual noncollinear planar vertices and erected equilateral-triangle centroids; inner construction may degenerate.

References

Euler Line Calculator — tool.io

Similar or alternative tools

Nine Point Circle Calculator
Construct nine-point circle and actual triangle side midpoints; center uses circumcenter/orthocenter and radius is half circumradius, not a generic circle from radius.
Napoleon Triangle Area Calculator
Compute outer/inner Napoleon triangle areas from actual noncollinear planar vertices and erected equilateral-triangle centroids; inner construction may degenerate.