Question

In triangle abc g is the centroid then ag2+bg2+cg2/ab2+bc2+ca2

Answer 1

Answer:

1/3

Step-by-step explanation:

Consider AG² + BG²+ CG²

If AD, BE and CF are the medians , we know that centroid divides the median in the ratio 2:1, thus

= (2/3*AD)² + (2/3*BE)² + (2/3*CF)²,

= 4/9[ AD² + BE² + CF²]

Now we know length of median formula  

Similarly BE² = AD² = 1/4*[2*(AB²+BC²)-AC²] , and

CF² = 1/4*[2*(BC²+AC²)-AB²]

Therefore,

AG² + BG²+ CG² = 4/9*1/4*3*(AB²+BC²+AC²)

=1/3*(AB²+BC²+AC²)

Thus, AG² + BG²+ CG²/AB²+BC²+AC² = 1/3.