`O A B C` is regular tetrahedron in which `D` is the circumcentre of ` O A B` and E is the midpoint of edge `A Cdot` Prove that `D E` is equal to half the edge of tetrahedron.

D , E ,a n dF are the middle points of the sides of the triangle A B C , then centroid of the triangle D E F is the same as that of A B C orthocenter of the tirangle D E F is the circumcentre of A B C orthocenter of the triangle D E F is the incenter of A B C centroid of the triangle D E F is not the same as that of A B Cdot

Let A B C be a triangle with A B=A Cdot If D is the midpoint of B C ,E is the foot of the perpendicular drawn from D to A C ,a n dF is the midpoint of D E , then prove that A F is perpendicular to B Edot

The mean of a, b, c, d and e is 28. If the mean of a, c and e is 24, then mean of b and d is

The lengths of two opposite edges of a tetrahedron are a and b ; the shortest distane between these edges is d , and the angel between them is theta Prove using vectors that the volume of the tetrahedron is (a b dsi ntheta)/6 .

Let O A C B be a parallelogram with O at the origin and O C a diagonal. Let D be the midpoint of O Adot using vector methods prove that B Da n dC O intersect in the same ratio. Determine this ratio.

Let O A C B be a parallelogram with O at the origin and O C a diagonal. Let D be the midpoint of O Adot using vector methods prove that B Da n dC O intersect in the same ratio. Determine this ratio.

If O A B C is a tetrahedron where O is the orogin anf A ,B ,a n dC are the other three vertices with position vectors, vec a , vec b ,a n d vec c respectively, then prove that the centre of the sphere circumscribing the tetrahedron is given by position vector (a^2( vec bxx vec c)+b^2( vec cxx vec a)+c^2( vec axx vec b))/(2[ vec a vec b vec c]) .

In a regular tetrahedron, prove that angle theta between any edge and the face not containing that edge is given by cos theta = 1/sqrt3 .

if vec Ao + vec O B = vec B O + vec O C ,than prove that B is the midpoint of AC.