Answer to Question #97211 in Analytic Geometry for Aravind

Question

What are the coordinates of the centroid of the triangle with vertices (X1,Y1), (X2,Y2), (X3 ,Y3)

Answer 1

Let ABC be the triangle where A(x1, y1), B(x2, y2) and C(x3, y3):

Let G be the centroid of triangle ABC.

Let AD be the median of triangle ABC. So, D is the midpoint of BC.

Midpoint of B(x2, y2) and C(x3, y3) is

$D= (\frac {x_2 +x_3} {2},\frac {y_2 +y_3} {2})$

We know that centroid divides median in the ratio 2:1. So, centroid G divides the median AD in the ratio 2:1.

We know that coordinate of point P(x, y) that divides the line segment joining  internally in the ratio m:n is

$P(x,y)= (\frac {mx_2 +nx_1} {m+n},\frac {my_2 +ny_1} {m+n})$

Here,

$x_1=x_1$ ,

$y_1=y_1$ ,

$x_2=\frac{x_2+x_3}{2}$ ,

$y_2=\frac{y_2+y_3}{2}$ ,

$m=2$ ,

$n=1$

Coordinate of G are

$G(x,y)= \frac {2 (\frac {x_2+x_3}{2}) +1(x_1)}{2+1},\frac {2 (\frac {y_2+y_3}{2}) +1(y_1)}{2+1} =(\frac {x_2+x_3+x_1}{3},\frac {y_2+y_3+y_1}{3})$

Hence Coordinate of centroid are

$G(x,y)=(\frac {x_1+x_2+x_3}{3},\frac {y_1+y_2+y_3}{3})$