Fermat Point Triangle Calculator

Euclidean triangle geometric median/Fermatpointactualxyvertices; vertex angle≥120°returns vertex, otherwise weighted-distance iteration within relative convergence. Non degenerate finite geometry, not trilinearidentityor length-only point.

Description

The Fermat point minimizes total Euclidean distance to the vertices of a nondegenerate triangle.

The Fermat point minimizes total Euclidean distance to the vertices of a nondegenerate triangle.^[1]

This tool returns a qualifying obtuse vertex or computes an interior geometric median with a bounded iteration.

When to use Fermat Point Triangle Calculator

This tool returns a qualifying obtuse vertex or computes an interior geometric median with a bounded iteration.

How to use Fermat Point Triangle Calculator and how it works

Prepare triangle coordinates, using the input definitions and matching units below.
Enter the required fields and run Fermat Point Triangle Calculator.
Read result.fermatPoint using the units, signs and ordering explained below.
Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.

Input field	Meaning	Value type	Requirement
actualTriangleCoordinates	Exactly three [x,y] vertices in matching Euclidean coordinate units.	array of array of number	Required; other constraints below

Require exactly three finite xy vertices. Translate and normalize by the largest distance from the first vertex and reject a normalized area determinant below 1e-12. If a vertex angle is at least 120 degrees, return that input vertex. Otherwise start at the normalized centroid and apply inverse-distance weighted Weiszfeld updates up to 10000 iterations; require movement below 1e-12, then transform back. Nonconvergence is an error. Return xy coordinates in the original units without a certified global numerical error bound.

point=argmin_x(sum(distance(x,vertex_i))); iteration=sum(vertex_i/distance_i)/sum(1/distance_i)

Apply the input ordering, units and branch conditions stated above.

Output field	Meaning	Value type
result	Object containing the calculated or selected fields below.	object
result.fermatPoint	Returned [x,y] geometric-median coordinates in the supplied coordinate units.	array of number

json

{
  "input": {
    "actualTriangleCoordinates": [
      [
        0,
        0
      ],
      [
        1,
        0
      ],
      [
        0.5,
        0.8660254037844386
      ]
    ]
  },
  "output": {
    "result": {
      "fermatPoint": [
        0.5,
        0.28867513459481287
      ]
    }
  }
}

Limitations and assumptions

The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.

All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.

Alternative or Complementary methods

Use Morley Triangle Calculator for a complementary operation: The first interior Morley triangle is formed by adjacent interior angle trisectors of a nondegenerate triangle.

Use Incircle Radius Calculator for a complementary operation: A triangle's inradius equals its area divided by its semiperimeter.

References

Fermat Point Triangle Calculator — tool.io

Similar or alternative tools

Morley Triangle Calculator
The first interior Morley triangle is formed by adjacent interior angle trisectors of a nondegenerate triangle.
Incircle Radius Calculator
A triangle's inradius equals its area divided by its semiperimeter.