Centroid Incentre and Orthocentre: Explained with Examples

A triangle has three famous "centres", and each one is found in a different way. The centroid is the point where the three medians meet; the incentre is the point where the three angle bisectors meet; and the orthocentre is the point where the three altitudes meet.

They matter because each centre has a special geometric job: the centroid is the triangle's balance point, the incentre is the centre of the circle that fits snugly inside the triangle, and the orthocentre anchors the triangle's altitudes. Architects, engineers and physicists use these points whenever triangles carry loads or define shapes.

Key Concepts

• Median: a line from a vertex to the midpoint of the opposite side. All three medians cross at the centroid.
• Angle bisector: a line that splits a vertex angle into two equal halves. All three meet at the incentre.
• Altitude: a perpendicular dropped from a vertex to the opposite side (extended if needed). All three meet at the orthocentre.
• Side lengths a, b, c: standard notation — side a = BC is opposite vertex A(x1, y1), side b = CA opposite B(x2, y2), side c = AB opposite C(x3, y3).
• Units: all three centres are points, so their coordinates use the same units as the vertices.

Formulas

For a triangle with vertices A(x1, y1), B(x2, y2), C(x3, y3):

• Centroid G = ( (x1+x2+x3)/3, (y1+y2+y3)/3 ) — the simple average of the three vertices. It divides each median in the ratio 2:1, vertex to midpoint.
• Incentre I = ( (ax1+bx2+cx3)/(a+b+c), (ay1+by2+cy3)/(a+b+c) ) — vertices weighted by the lengths of the opposite sides.
• Orthocentre H: found by writing the equations of any two altitudes (lines through a vertex perpendicular to the opposite side) and solving them together; there is no single short average formula.

Assumptions: the three vertices are not collinear (otherwise there is no triangle). Limits: for an obtuse triangle the orthocentre falls outside the triangle, and for a right-angled triangle it sits exactly on the right-angle vertex.

Worked Example

Find the centroid of the triangle with vertices A(0, 0), B(6, 0) and C(3, 9).

1. Add the x-coordinates: 0 + 6 + 3 = 9.
2. Divide by 3: 9/3 = 3, so G has x = 3.
3. Add the y-coordinates: 0 + 0 + 9 = 9.
4. Divide by 3: 9/3 = 3, so G has y = 3.
5. Therefore the centroid is G = (3, 3).
6. Sanity check: the median from C(3, 9) goes to the midpoint of AB, which is (3, 0). The point (3, 3) lies on this vertical median and is twice as far from C as from the midpoint (distances 6 and 3), matching the 2:1 rule.

Common Misconceptions

• "All three centres are the same point." They coincide only in an equilateral triangle. In a scalene triangle all three are different points.
• "The orthocentre is always inside the triangle." In an obtuse triangle it lies outside; only acute triangles keep it inside.
• "The incentre formula is a simple average." The centroid is the simple average, but the incentre weights each vertex by its opposite side length — longer sides pull the incentre toward themselves.

Real-World Applications

• The centroid is the centre of mass of a uniform triangular plate — engineers balance triangular parts on it.
• The incentre is the centre of the largest circle that fits inside a triangular plot or machine part (the incircle).
• Architects use the orthocentre when laying out perpendicular supports from triangle vertices.
• Computer graphics engines compute centroids to position and rotate triangular meshes.

Centroid, incentre and orthocentre are the three meeting points of a triangle's medians, angle bisectors and altitudes. The centroid is a plain average, the incentre is a side-weighted average, and the orthocentre comes from intersecting altitudes — three centres, three different jobs.