How to use the triangle centers calculator
- Enter the x and y coordinates of the three corners A, B and C, in any one unit.
- Read the circumcenter in the headline, then the full list of centers in the table.
- Check the circumradius, inradius, area, side lengths and angles in the results.
- Use the steps to see how each center is computed and the Euler line check.
Formula
Centroid G = (A + B + C) / 3
Circumcenter O from perpendicular bisectors; R = a b c / (4 x area)
Incenter I = (a A + b B + c C) / (a + b + c); r = area / s, s = half the perimeter
Orthocenter H = A + B + C - 2 O
Nine-point center N = (O + H) / 2, with radius R / 2
a, b and c are the lengths of the sides opposite A, B and C. The Euler line passes through O, G and H, with G one third of the way from O to H, and the distance OI satisfies OI squared = R (R - 2r).
A scalene triangle
Take A (0, 0), B (6, 0) and C (2, 5). The area is 15, the sides are a = 6.4031, b = 5.3852 and c = 6, and the angles at A, B and C are 68.20, 51.34 and 60.46 degrees.
The centroid is (2.6667, 1.6667), the circumcenter (3, 1.7) with R = 3.4482, the incenter (2.4910, 1.6865) with r = 1.6865, the orthocenter (2, 1.6) and the nine-point center (2.5, 1.65). The circumcenter, centroid and orthocenter lie on one line, as the Euler line check confirms.
The five special points
A triangle has several centers, each defined by a different meeting of lines. The centroid is where the three medians meet and is the balance point of a flat triangle. The circumcenter is the center of the circle through all three corners. The incenter is the center of the largest circle inside the triangle, touching all three sides. The orthocenter is where the altitudes meet, and the nine-point center is the center of a circle that passes through nine named points of the triangle.
For an equilateral triangle all five coincide. In a right triangle the circumcenter is the midpoint of the longest side and the orthocenter is the right-angle corner. In an obtuse triangle the circumcenter and orthocenter fall outside.
Using coordinates
Starting from coordinates avoids having to build the triangle from sides and angles. If you only know the three sides, the triangle calculator finds angles and area first, and you can place the corners on an axis to get coordinates. The circle from three points calculator gives just the circumcenter and radius.
The polygon area from coordinates calculator uses the same shoelace formula for the area used here, and the Pythagorean theorem calculator covers the right triangle case.
Assumptions and limits
- The corners are flat x and y coordinates in one unit; the three points must not lie on a line.
- All distances and areas are in that unit and its square.
- Results are rounded for display to four decimals but computed in full precision.
- The Euler line check is a numerical test and may show tiny differences from rounding.
Frequently asked questions
Which center is the center of gravity of the triangle?
The centroid. For a thin flat triangle of uniform material it is the balance point, and it is always inside the triangle, two thirds of the way along each median from the corner.
Why can the circumcenter be outside the triangle?
In an obtuse triangle the perpendicular bisectors meet beyond the longest side. The circle through the three corners is then centered outside the triangle, on the far side of that longest side.
What is the Euler line?
It is the line through the circumcenter, centroid and orthocenter (and the nine-point center). The centroid is always twice as far from the orthocenter as from the circumcenter. For an equilateral triangle the three points merge, so the line is not defined.
Can I enter the sides instead of coordinates?
Not directly. Place one corner at the origin and another on the x axis at the length of one side, then find the third corner from the other two sides. Alternatively, use the triangle calculator to get angles and area.