Equilateral and isosceles triangles have symmetry, and this symmetry can be used to find the centre of the triangle. The centre of an equilateral triangle is quickly found to be two thirds of the distance from the vertex to the centre of the the opposite side, by a simple rotation argument,
and the same result can be proved for an isosceles triangle by integration, taking moments about the odd side.
For a scalene triangle no symmetry argument is possible, but a vector argument can be made. Consider the triangle ABC below.
When a line is drawn through a vertex to the opposite side through the centre of the triangle, half the area of the triangle must lie on either side, so the line drawn must bisect that side. Draw lines from A and C to the centres of the opposite sides, N and M respectively. These lines must intersect at the centre of the triangle, G below.
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And since G is on
but
so
and
Taking C as the origin, the position vector of G is
or
Equating these gives
Now equate coefficients of
Equate coefficients of
Equating these simultaneously gives
and the centre of the triangle is two thirds of the way from any vertex to the centre of the opposite side.