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,` where `P` and `Q` are middle points of diagonals `A C` and `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

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 convex quadrilateral A B C D ,A B=a ,B C=b ,C D=c ,D A=d . This quadrilateral is such that a circle can be inscribed in it and a circle can also be circumscribed about it. Prove that tan^2A/2=(b c)/(a d)dot

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 .

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)

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

In a scalene triangle A B C ,D is a point on the side A B such that C D^2=A D D B , sin s in A S in B=sin^2C/2 then prove that CD is internal bisector of /_Cdot

A quadrilateral A B C D is such that diagonal B D divides its area in two equal parts. Prove that B D bisects A Cdot GIVEN : A quadrilateral A B C D in which diagonal B D bisects it. i.e. a r( A B D)=a r( B D C) CONSTRUCTION : Join A C Suppose A C and B D intersect at O . Draw A L_|_B D and C M B Ddot TO PROVE : A O=O Cdot