`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.

O is the circumcentre of triangleABC and D is the midpoint of BC, if angleBAC = 40^@ , then angleBOD =?

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 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.

If O A B C is a tetrahedron where O is the origin and 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]) .

The chord P Q of the rectangular hyperbola x y=a^2 meets the axis of x at A ; C is the midpoint of P Q ; and O is the origin. Then A C O is (a) equilateral (b) isosceles (c) right-angled (d) right isosceles

Statement 1:Let A( vec a),B( vec b)a n dC( vec c) be three points such that vec a=2 hat i+ hat k , vec b=3 hat i- hat j+3 hat ka n d vec c=- hat i+7 hat j-5 hat kdot Then O A B C is a tetrahedron. Statement 2: Let A( vec a),B( vec b)a n dC( vec c) be three points such that vectors vec a , vec ba n d vec c are non-coplanar. Then O A B C is a tetrahedron where O is the origin.

A point O is the centre of a circle circumscribed about a triangle A B Cdot Then vec O Asin2A+ vec O Bsin2B+ vec O Csin2C is equal to

In triangle A B C ,/_A=30^0,H is the orthocenter and D is the midpoint of B C . Segment H D is produced to T such that H D=D T The length A T is equal to (a). 2B C (b). 3B C (c). 4/2B C (d). none of these

In A B C Prove that A B^2+A C^2=2(A O^2+B O^2) , where O is the middle point of B C