If L,M,N divide the sides BC,CA and AB of a triangle ABC in the same ratio, then show that the traiangle ABC and dLMN have the same centroid.

n_(1),n_(2)andn_(3) points are given on the sides BC ,CA and AB respectively of the triangle ABC . Find the number of triangles formed by taking these given points as vertices of a triangle .

The ratio of the sides of a triangle ABC is 1: sqrt3:2. Then the ratio A:B:C is-

D, E, F are mid points of sides BC, CA, AB of Delta ABC . Find the ratio of areas of Delta DEF and Delta ABC .

The sides AB, BC and CA of a triangle ABC have 3,4 and 5 interior points respectively on them . the number of triangle that can be constructed using these interior points as vertices , is

The cordinates of the mid points of the sides BC, CA and AB of the triangle ABC are (5,2,8), (2,-2,-3) and (2,-3,4) , find the coordinates of the centroid of the triangle.

D, E, F are the midpoints of the sides bar(BC), bar(CA) " and " bar(AB) respectively of the triangle ABC. If P is any point in the plane of the triangle, show that vec(PA)+vec(PB)+vec(PC)=vec(PD)+vec(PE)+vec(PF) .

If the lengths of the sides vec(BC), vec(CA) and vec(AB) of the triangle ABC be a:b:c respectively and the lengths of perpendicular from the circumcenter upon vec(BC), vec(CA) and vec(AB) be x, y, z respectively, then prove that, a/x + b/y + c/z = (abc)/(4xyz) .

If the medium AD is perpendicular to the side AB in the triangle ABC, show that tan A+ 2 tan B=0 .

G is a point inside the plane of the triangle ABC, if vec(GA)+vec(GB)+vec(GC)=vec(0) , then show that G is the centroid of the triangle ABC.