prove the coordinate of the centroid of triangle ABC is ordered pair (x1 + x2+ x3)/3, (y1 + y2 + y3)/3?

Question

given the triangle ABC with A (x1,y1), B(x2,y2), C (x3,y3).
write out full proof please with numbered statements and reasons.

note: centroid is the point of intersection of 2 medians in a triangle.

Answer 1

median wud join A to mid pt of BC.
mid pt of BC = (x2 + x3)/2 , (y2 + y3)/2

like this v hav 2 medians. (x1,y1) to ( (x2+x3)/2 , (y2+y3)/2 )
and (x2,y2) to ( (x1+x3)/2 , (y1+y3)/2 )


now 2 medians intersect in d ratio 2:1.

by section formula: 2 get pt of intersetion;



[ 2 {(x2+x3)/2} + x1 ] / (2 + 1) and [ 2 {(Y2+y3)/2} + y1 ] / (2 + 1)
i.e

(x1 + x2+ x3)/3, (y1 + y2 + y3)/3