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.
Description
The nine-point circle passes through a triangle's side midpoints and related altitude points.
The nine-point circle passes through a triangle's side midpoints and related altitude points. Its center lies halfway between circumcenter and orthocenter, with half the circumradius.1
Construct nine-point circle and actual triangle side midpoints; center uses circumcenter/orthocenter and radius is half circumradius, not a generic circle from radius.
When to use Nine Point Circle Calculator
Use this tool to construct nine-point circle and actual triangle side midpoints; center uses circumcenter/orthocenter and radius is half circumradius, not a generic circle from radius.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Nine Point Circle 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 ninePointCircleCenter, ninePointCircleRadius, sideMidpoints. 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 exactly three planar coordinate pairs in consistent units. Reject collinear or poorly conditioned triangles, calculate the two centers, then return circle center, radius and cyclic side midpoints.
nine-point center = (circumcenter+orthocenter)/2; radius = circumradius/2 Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| ninePointCircleCenter | Nine point circle center | list of number |
| ninePointCircleRadius | Nine point circle radius | number |
| sideMidpoints | Side midpoints | list of list of number |
{
"input": {
"vertices": [
[
0,
0
],
[
4,
0
],
[
0,
3
]
]
},
"output": {
"ninePointCircleCenter": [
1,
0.75
],
"ninePointCircleRadius": 1.25,
"sideMidpoints": [
[
2,
0
],
[
2,
1.5
],
[
0,
1.5
]
]
}
}Limitations and assumptions
Nearly collinear inputs are rejected at the helper's relative 10⁻¹² area-conditioning threshold. This does not construct all nine points.
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.
Alternative or Complementary methods
Use Euler Line Calculator: Construct circumcenter, centroid, orthocenter and Euler-line direction from planar vertices; equilateral coincident centers leave no unique Euler line. 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
-
Nine Point Circle Calculator — tool.io
Similar or alternative tools
- Euler Line Calculator
Construct circumcenter, centroid, orthocenter and Euler-line direction from planar vertices; equilateral coincident centers leave no unique Euler line.
- Napoleon Triangle Area Calculator
Compute outer/inner Napoleon triangle areas from actual noncollinear planar vertices and erected equilateral-triangle centroids; inner construction may degenerate.