IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry - Straight Line [Admin]:centroid of triang., Proof

Akshay

Ref: IIT Mathematics M.L Khanna pg no.5, 2007 edition

If G be the centroid of triangle ABC and O be any other point then prove that

1)AB²+BC²+CA² = 3(GA²+GB²+GC²)

2)OA²+OB²+OC²=GA²+GB²+GC²+3GO²

gopi_mith

Without loss of generality, u can assume that the centroid is at the origin and then solve by applying distance formula

edison

Consider a triangle ABC with centroid at G and the coordinaes be

A(0,0); B(x₁,y₁); C(x₂,y₂)

then coordinates of point 'G' the centroid of the triangle is given by

x = (0 + x₁ + x₂) /3 = (x₁ + x₂) /3

y = (0 + y₁ + y₂) /3 = (y₁ + y₂) /3

Now to prove

AB²+BC²+CA² = 3(GA²+GB²+GC²)

Substitute the coordinates at each sides and simplify to obtain above result.

Similarly take O(x,y) to prove second result