Prove that if z1 z2 z3 are noncollinear points in the comple

Prove that if z_1, z_2, z_3 are non-collinear points in the complex plane, the medians of the triangle with vertices z_1, z_2, z_3 intersect at the point 1/3(z_1 + z_2 + z_3). What geometric fact does this prove?

Solution

given z1,z2,z3 are 3 noncollinear points in a complex plane, the medians of the triangle intersect at the point 1/3(z1+z2+z3)

This comes from the fact that

If (x1,y1),(x2,y2),(x3,y3) are three noncollinear points

then the point of intersection of medians is called centroid of the triangle

The centroid of the triangle is given by

((x1+x2+x3)/3,(y1+y2y+y2)/3)

(1/3(x1+x2x+x3),1/3(y1+y2+y3))