In a quadrilateral `A B C D` prove that `A B^2+B C^2+C D^2+D A^2=A C^2+B D^2+4P Q^2\ w h e r e\ P\ a n d\ Q` are middle points of diagonals `A C\ a n d\ B Ddot`

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

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

B D is one of the diagonals of a quadrilateral A B C Ddot\ \ A M\ a n d\ C N are the perpendiculars from A\ a n d\ C , respectively, on B Ddot Show That a r\ (q u a ddotA B C D)=1/2B Ddot(A M+C N)

Prove that the figure formed by joining the mid-points of the pair of consecutive sides of a quadrilateral is a parallelogram. OR B A C D is a parallelogram in which P ,Q ,R and S are mid-points of the sides A B ,B C ,C D and D A respectively. A C is a diagonal. Show that : P Q|| A C and P Q=1/2A C S R || A C and S R=1/2A C P Q=S R (iv) P Q R S is a parallelogram.

In Figure, B E_|_A C ,\ A D is any line from A\ to\ B C intersecting B E in HdotP ,\ Q\ a n d\ R are respectively the mid-points of A H ,\ A B\ a n d\ B Cdot Prove that /_P Q R=90^0

The diagonals of quadrilateral A B C D ,A C and B D intersect in Odot Prove that if B O=O D , the triangles A B C and A D C are equal in area. GIVEN : A quadrilateral A B C E in which its diagonals A C and B D intersect at O such that B O = O Ddot TO PROVE : a r( A B C)=a r( A D C)

In Figure, diagonal A C of a quadrilateral A B C D bisects the angles A\ a n d\ C . Prove that A B=A D\ a n d\ C B=C D

A B C D is a parallelogram, G is the point on A B such that A G=2\ G B ,\ E\ is a point of D C such that C E=2D E\ a n d\ F is the point of B C such that B F=2F Cdot Prove that: a r\ (\ E B G)=a r\ (\ E F C)

A B C D is a quadrilateral. A line through D , parallel to A C , meets B C produced in P as shown in Figure. Prove that a r\ ( A B P)=a r\ (Q u a ddotA B C D)dot