The mid-point of the sides of a triangle are (1, 5, -1), (0, 4, -2) and (2, 3, 4) find its vertices and also find the centroid of the triangle.

Are the points A(3,6,9), B(10,20,30) and C(25,-41,5), the vertices of a right angled triangle?

Prove that the coordinates of the points which divide the lines joining the vertices of a tetrahedron to the centroid of the opposite faces in the ration 3:1 are same.

Using the section formula, show that the points, (2, -3, 4), (-1, 2, 1) and (0, \(\frac{1}{3}\), 2) are collinear.

Three points A(3, 2, 0), B(5, 3, 2) and C(-9, 6, -3) are forming a triangle . The bisector Ad of <BAC meets BC at  D. Find the coordinates of D

Find a point in XY plane which is equidistant from three points (2, 0, 3), (0, 3, 2) and (0, 0, 1).

Find the locus of the point which is equidistant from A(3, 4, 0) and B(5, 2, -3).

Given that P(5, 4, -2), Q (7, 6, -4) and R (11, 10, -8) are collinear points. Find the ratio in which Q divides PR.

If the origin is the centroid of the triangle with vertices A (3a, 4, - 5), B (- 2, 4b, 6), C (6, 10,c). Find the value of a, b, c.

Show that the coordinates of the centroid of a triangle with vertices A(x₁,x ₂,x₃)., b(y₁,y ₂,y ₃), c(z₁,z₂,z ₃) are \(\left[ \frac { x1+x2+x3 }{ 3 } ,\frac { y1+y2+y3 }{ 3 } ,\frac { z1+z2+z3 }{ 3 } \right] \)

Find the coordinates of a point on y-axis which are at a distance of \(5\sqrt { 2 } \)  from the point P(3, -2,5).