Triangle Centres Calculator
Find the area, centroid, circumcentre, orthocentre and incentre of a triangle from its three vertices, with exact fractions and the Euler line.
About the Triangle Centres Calculator
The centroid G is where the medians meet; the circumcentre O is equally far from the three vertices; the orthocentre H is where the altitudes meet; the incentre I is where the angle bisectors meet. O, G and H always lie on one line (the Euler line), with G a third of the way from O to H.
The area uses the determinant formula ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.
The formula
Area = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|, G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)
- Incentre
- (aA + bB + cC)/(a + b + c), a = |BC| etc.
- Euler line
- H = 3G − 2O
Worked example
A(0, 0), B(6, 0), C(2, 4)
- Area = ½|0 + 6 × 4 + 0| = 12
- G = (8/3, 4/3), O = (3, 1), H = (2, 2)
Questions people ask
Which centre is the centre of gravity?
The centroid G: a flat triangle of uniform card balances on it.
Why is O outside my triangle?
In an obtuse triangle the circumcentre and orthocentre lie outside it; in a right triangle O is the midpoint of the hypotenuse and H is the right-angled vertex.