The formula
centroid = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
Where it comes from
The centroid is the plain average of the three vertices, and it is where the triangle balances. The incentre is the same idea weighted by the side lengths, and it is the centre of the circle that fits inside touching all three sides.
How to work it out by hand
- Centroid: average the three x values and the three y values
- Work out the three side lengths from the coordinates
- Incentre: average the vertices weighted by the length of the opposite side
- Area from the shoelace formula on the coordinates
What is worth knowing
A triangle has four classical centres and they are all different in general: centroid, incentre, circumcentre and orthocentre. Three of them — everything except the incentre — always lie on a single straight line, the Euler line, which Euler proved in 1765 and which is still surprising. The centroid divides each median in a 2:1 ratio, closer to the side than the vertex, and it is the only one of the four that is guaranteed to be inside the triangle along with the incentre. Physically it is the centre of mass of a uniform triangular plate, which is what 'balances' means here.