If `A D` is the median of ` A B C ,` using vectors, prove that `A B^2+A C^2=2(A D^2+C D^2)dot`

In A B C , A D is a median. Prove that A B^2+A C^2=2\ A D^2+2\ D C^2 .

A point D is taken on the side B C of a A B C such that B D=2d Cdot Prove that a r( A B D)=2a r( A D C)dot

In figure, AD is a median of a triangle ABC and A M_|_B C . Prove that: (i) A C^2=A D^2+B C.D M+ ((B C)/2)^2 (ii) A B^2=A D^2-B CdotD M+ ((B C)/2)^2 (iii) A C^2+A B^2=2A D^2+ 1/2B C^2

If A D is a median of a triangle A B C ,\ then prove that triangles A D B\ a n d\ A D C are equal in area. If G is the mid-point of median A D , prove that a r\ ( B G C)=2a r\ (\ A G C)

In a quadrilateral A B C D , given that /_A+/_D=90o . Prove that A C^2+B D^2=A D^2+B C^2 .

In figure, ABC is a triangle in which /_A B C=90^@ and A D_|_B C . Prove that A C^2=A B^2+B C^2-2B C.B D .

If AD, BE and CF are the medians of a DeltaA B C ,t h e n(A D^2+B E^2+C F^2):(B C^2+C A^2+A B^2) is equal to

The A D ,B Ea n dC F are the medians of a c A B C , then (A D^2+B E^2+C F^2):(B C^2+C A^2+A B^2) is equal to

In figure, ABC is triangle in which /_A B C > 90o and A D_|_C B produced. Prove that A C^2=A B^2+B C^2+2B C.B D .

In an equilateral triangle A B C if A D_|_B C , then A D^2 = (a)C D^2 (b) 2C D^2 (c) 3C D^2 (d) 4C D^2