BYTETOOLS

Triangle Centers Calculator

Enter three vertex coordinates to locate the centroid, circumcenter, incenter and orthocenter of a triangle, with the circumradius, inradius and Euler line.

6
Area
12
Perimeter
2.5
Circumradius R
1
Inradius r

The four classical centers

CenterCoordinatesHow it is found
Centroid(1.333333, 1)Average of the three vertices
Circumcenter(2, 1.5)Where the perpendicular bisectors meet
Incenter(1, 1)Side-length weighted average of the vertices
Orthocenter(0, 0)Where the three altitudes meet

The centroid, circumcenter and orthocenter always lie on one straight line — the Euler line — with the centroid exactly one third of the way from the circumcenter to the orthocenter.

Sketch

CentroidCircumcenterIncenterOrthocenter

Green centroid, blue circumcenter, amber incenter, rose orthocenter. The dashed circles are the circumcircle and incircle; the long dashed line is the Euler line.

Side lengths

  • a (opposite A, from B to C) = 5
  • b (opposite B, from C to A) = 3
  • c (opposite C, from A to B) = 4
  • Semi-perimeter s = 6
  • Circumradius R = abc / 4·Area = 2.5
  • Inradius r = Area / s = 1

What is the Triangle Centers Calculator?

The ByteTools Triangle Centers Calculator takes three vertex coordinates and finds the four classical centers at once: the centroid, the circumcenter where the perpendicular bisectors meet, the incenter as the side-length weighted average of the vertices, and the orthocenter where the three altitudes cross.

  • Centroid, circumcenter, incenter and orthocenter from three coordinates
  • Circumradius from abc/4A and inradius from area over semi-perimeter
  • Euler line drawn through the centroid, circumcenter and orthocenter
  • Colour-coded sketch with the circumcircle and incircle
  • Detects collinear or repeated points instead of returning nonsense
  • Everything computed locally in your browser

How to use the Triangle Centers Calculator

  1. 1

    Type the x and y coordinates of vertex A, vertex B and vertex C.

  2. 2

    Read the area, perimeter, circumradius and inradius in the summary tiles.

  3. 3

    Check the table for the coordinates of all four centers and how each one is found.

  4. 4

    Use the sketch to see the centers, the circumcircle, the incircle and the Euler line together.

  5. 5

    Click Copy centers to grab the coordinates.

About the Triangle Centers Calculator

The ByteTools Triangle Centers Calculator takes three vertex coordinates and finds the four classical centers at once: the centroid, the circumcenter where the perpendicular bisectors meet, the incenter as the side-length weighted average of the vertices, and the orthocenter where the three altitudes cross.

Alongside the coordinates you get the triangle's area, perimeter, all three side lengths, the circumradius R = abc/4A and the inradius r = A/s. A scale drawing plots the triangle with each center colour-coded, the circumcircle and incircle dashed in, and the Euler line running through the centroid, circumcenter and orthocenter.

The geometry is computed in your browser with JavaScript — nothing is uploaded, so the tool is private and works offline. Collinear or repeated points are caught and explained, since three points on a line do not form a triangle.

Frequently asked questions

What is the difference between the centroid and the circumcenter?

The centroid is the average of the three vertices and is always inside the triangle — it is the balance point. The circumcenter is equidistant from all three vertices and can fall outside the triangle when one angle is obtuse. They only coincide in an equilateral triangle.

How do you find the orthocenter of a triangle?

Intersect any two altitudes, or use Euler's shortcut: H = A + B + C - 2O, where O is the circumcenter. For a right triangle the orthocenter is simply the right-angle vertex, which is a quick way to sanity-check the result.

What is the Euler line?

It is the straight line through the orthocenter, centroid and circumcenter of any non-equilateral triangle. The centroid always sits exactly one third of the way from the circumcenter to the orthocenter. In an equilateral triangle all three points coincide, so there is no distinct line to draw.

How is the incenter calculated from coordinates?

It is the weighted average of the vertices, using each opposite side length as the weight: (a·A + b·B + c·C)/(a + b + c). For a 3-4-5 triangle at (0,0), (4,0) and (0,3), that gives the incenter (1, 1) with an inradius of 1.

Why does the calculator refuse three points in a line?

Collinear points enclose zero area, so there is no circumcircle, no incircle and no unique center to report — the formulas would divide by zero. The tool detects the zero cross product and asks you to move a point instead of printing a meaningless answer.

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