Triangle Centers Solver (Incenter / Circumcenter / Centroid / Orthocenter)
Enter the three vertices A, B, C of a triangle to get the incenter, circumcenter, centroid and orthocenter coordinates, plus side lengths, area and a chart.
Input Data
Results
Triangle and the four centers
Chart notes:The chart shows the triangle and the four centers: for an equilateral triangle they coincide; for a right triangle the orthocenter is at the right angle and the circumcenter at the hypotenuse midpoint.
At a glance:The four triangle centers are the centroid G (intersection of medians, coordinate average), incenter I (intersection of angle bisectors, side-length weighted), circumcenter O (intersection of perpendicular bisectors, circumcircle center) and orthocenter H (intersection of altitudes). They are collinear on the Euler line with H = 3G − 2O.
Formula
Centroid G = ((xA+xB+xC)/3, (yA+yB+yC)/3).
Incenter I = (a·A + b·B + c·C)/(a+b+c), with a=|BC|, b=|CA|, c=|AB|.
Circumcenter O: intersection of perpendicular bisectors; D = 2[xA(yB−yC)+xB(yC−yA)+xC(yA−yB)].
Orthocenter H = 3G − 2O (Euler line).
Area: shoelace |xA(yB−yC)+xB(yC−yA)+xC(yA−yB)| / 2.
$$G = \left(\frac{x_A+x_B+x_C}{3},\;\frac{y_A+y_B+y_C}{3}\right)$$$$I = \frac{aA + bB + cC}{a+b+c}$$$$H = 3G - 2O$$How to Use
- Enter the x, y coordinates of the three vertices A, B, C.
- The tool instantly returns the centroid G, incenter I, circumcenter O and orthocenter H.
- It also gives side lengths a, b, c and the area, summarizing the relationship in the note.
- Use the case studies and the chart to verify your geometry working visually.
Common triangle centers (coordinates to 4 dp)
| Triangle | Centroid G | Incenter I | Circumcenter O | Orthocenter H |
|---|---|---|---|---|
| Equilateral A(0,0)B(2,0)C(1,√3) | (1, 0.5774) | (1, 0.5774) | (1, 0.5774) | (1, 0.5774) |
| Right A(0,0)B(3,0)C(0,4) | (1, 1.3333) | (1, 1) | (1.5, 2) | (0, 0) |
| Isosceles A(0,0)B(4,0)C(2,3) | (2, 1) | (2, 1) | (2, 0.75) | (2, 1.5) |
Equilateral: all four centers coincide. Right: orthocenter at right angle, circumcenter at hypotenuse midpoint. Obtuse: orthocenter and circumcenter outside.
Case Studies
Equilateral A(0,0), B(2,0), C(1,√3)
All three sides equal 2 — an equilateral triangle.
The four centers coincide at (1, 0.5774): G = I = O = H.
Area = √3 ≈ 1.7321, checking ½·2·√3 = √3.
Right triangle A(0,0), B(3,0), C(0,4)
Sides a=|BC|=5, b=|CA|=4, c=|AB|=3 (a 3-4-5 right triangle).
Centroid G=(1, 1.3333); circumcenter O is the midpoint of the hypotenuse (1.5, 2).
Orthocenter H = the right-angle vertex A = (0,0), since the two legs are perpendicular.
Isosceles right A(0,0), B(2,0), C(0,2)
Sides a=2√2≈2.8284, b=2, c=2.
Centroid G=(0.6667, 0.6667), circumcenter O=(1,1).
Orthocenter H=(0,0)=A, area = 2.
Obtuse A(0,0), B(5,0), C(2,2)
Sides a≈3.6056, b≈2.8284, c=5.
Centroid G=(2.3333, 0.6667).
Orthocenter H and circumcenter O both fall outside the triangle (typical of an obtuse triangle).
General A(1,1), B(4,5), C(7,2)
Sides a≈4.2426, b≈7.2111, c=5.
Centroid G=(4, 2.6667).
Incenter I (side-weighted) lies inside; circumcenter O from perpendicular bisectors.
Isosceles A(0,0), B(4,0), C(2,3)
Sides a=b≈3.6056, c=4, axis of symmetry x=2.
G, I, O, H all lie on the symmetry axis x=2.
Area = 6, checking ½·4·3 = 6.
FAQ
Are all four centers always inside the triangle?
No. The centroid and incenter are always inside. The circumcenter is inside for acute, at the hypotenuse midpoint for right, and outside for obtuse triangles. The orthocenter is inside for acute, at the right angle for right, and outside for obtuse.
What is the Euler line?
The centroid G, circumcenter O and orthocenter H (and the nine-point center) are collinear on the Euler line, with the vector relation H = 3G − 2O, i.e. OG : GH = 1 : 2.
Why is the incenter weighted by side lengths?
The incenter is the intersection of the internal angle bisectors and is equidistant from the three sides. Its coordinates are the side-length-weighted average of the vertices, equivalent to the angle bisector theorem.
How is the circumcenter related to the circumcircle?
The circumcenter O is the center of the circumcircle and is equidistant from all three vertices (the circumradius R). For a right triangle R = hypotenuse/2 and O is the hypotenuse midpoint.
Why can the orthocenter be outside?
In an obtuse triangle two of the three altitudes fall outside the figure, so their extensions meet outside — that is the orthocenter. This is typical of obtuse triangles.
How does the shoelace formula work?
Order the vertices clockwise or counter-clockwise; area = |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)| / 2. The tool applies it automatically and agrees with base×height÷2.
Any special property of isosceles triangles?
All four centers of an isosceles triangle lie on the perpendicular symmetry axis of the base, which is handy for sketching and verification with this tool.
Does the vertex order matter?
No. The tool is insensitive to the labeling order A, B, C; you get the same set of four centers (though the side labels a, b, c follow the chosen labeling).
Related Tools
References
Content review: Calculatorism Science Team. Centroid, incenter, circumcenter, orthocenter formulas and the Euler-line relation H = 3G − 2O have been verified. Results are for reference only.