Question

If O be the circum center; G, the centroid and H, the orthocenter of triangle ABC, prove that O, G, H are collinear and G divides OH in the ratio 1:2 

Answer 1

Consider position vector of A, B, C be taken as vector (a, b, c). And then use geometry of triangle to solve this problem.

Let O, the circumcenter of the ∆ ABC be chosen as origin and position vector of A, B, C be taken vector (a, b, c).

Hence position vector of G the centroid is

Let P be the point whose position vector is

In similar manner we can show that BP is perpendicular to AC and AP is perpendicular to CB. 

Hence P is the orthocentre which is H.

Above show that O, G, H are collinear and G divides OH in the ratio 1:2