If P, Q, R divide the sides BC, CA and AB of `DeltaABC` in the same ratio, prove that the centroid of the triangles `ABC and PQR` coincide.

Let a,b,c be the sides BC, CA, AB of DeltaABC on xy plane. If abscissa and ordinate of vertices of the triangle are integers and R is the circumradius, then 2R can be equal to

D, E and F are the mid-points of the sides BC, CA and AB respectively of Delta ABC and G is the centroid of the triangle, then vec(GD) + vec(GE) + vec(GF) =

Points L, M, N divide the sides BC, CA, AB of ABC in the ratio 1:4, 3:2, 3:7 respectively. Prove thatAL + BM + CŃ is a vector parallel to CK where K divides AB in the ratio 1: 3.

The vertices of a triangle an A(1,2), B(-1,3) and C(3, 4). Let D, E, F divide BC, CA, AB respectively in the same ratio. Statement I : The centroid of triangle DEF is (1, 3). Statement II : The triangle ABC and DEF have the same centroid.

If P, Q , R are the mid-points of the sides AB, BC and CA of Delta ABC and O is point whithin the triangle, then vec (OA) + vec(OB) + vec( OC) =

A (-4, 2), B(0, 2) and C(-2,-4) are vertices of a triangle ABC. P, Q and R are mid-points of sides BC, CA and AB respectively. Show that the centroid of Delta PQR is the same as the centroid of Delta ABC.

The sides AB and BC and the median AD of triangle ABC are equal to the sides PQ and QR and the median PM of triangle PQR respectively. Prove that the triangles ABC and PQR are congruent.

If D(-2, 3), E (4, -3) and F (4, 5) are the mid-points of the sides BC, CA and AB of the sides BC, CA and AB of triangle ABC, then find sqrt((|AG|^(2)+|BG|^(2)-|CG|^(2))) where, G is the centroid of Delta ABC .

Normal at 3 points P, Q, and R on a rectangular hyperbola intersect at a point on a curve the centroid of triangle PQR

In the adjoining figure D, E and F are the mid-points of the sides BC, CA and AB of the equilateral DeltaABC. Prove that DeltaDEF is also an equilateral triangle.