If P be any point in the plane of square ABCD, prove that
`PA^(2)+PC^(2)=PB^(2)+PD^(2)`

ABC is a right triangle right angled at B. Let D and E be any points on AB and BC respectively. Prove that AE^(2) + CD^(2) = AC^(2) + DE^(2) .

lf G be the centroid of a triangle ABC and P be any other point in the plane prove that PA^2+PB^2+PC^2=GA^2+GB^2+GC^2+3GP^2

In DeltaABC, /_B=90^(@) . D and E are any points on sides AB and BC respectively. Prove that AE^(2)+CD^(2)= AC^(2)+DE^(2) .

Let S be the square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c and d denote the lengths of the sides of the quadrilateral, prove that 2 le a^(2)+b^(2)+c^(2)+d^(2)le 4 .

For any triangle ABC, prove that : a(bcosC-c cosB)=b^(2)-c^(2)

For any triangle ABC, prove that : (sin(B-C))/(sin(B+C))=(b^(2)-c^(2))/(a^(2))

If A and B be the points (3, 4, 5) and (-1, 3, -7) respectively, find the equation of the set of points P such that PA^(2)+PB^(2)=k^(2) , where k is a constant.

If P be a point on the plane lx+my+nz=p and Q be a point on the OP such that OP. OQ=p^2 show that the locus of the point Q is p(lx+my+nz)=x^2+y^2+z^2 .

O is any point inside a rectangle ABCD (see the given figure). Prove that OB^(2)+OD^(2)= OA^(2)+OC^(2) .