If 'O' is any point in the interior of rectangle ABCD, then prove that : `OB^(2) + OD^(2) = OA^(2) + OC^(2)`

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

In adjoining figure, point T is in the interior of rectangle PQRS. Prove that, TS^(2) + TQ^(2) = TP^(2) + TR^(2)

If A^(3)=O , then prove that (I-A)^(-1) =I+A+A^(2) .

If O( vec0 ) is the circumcentre and O' the orthocentre of a triangle ABC, then prove that i. vec(OA)+vec(OB)+vec(OC)=vec(OO') ii. vec(O'A)+vec(O'B)+vec(O'C)=2vec(O'O) iii. vec(AO')+vec(O'B)+vec(O'C)=2 vec(AO)=vec(AP) where AP is the diameter through A of the circumcircle.

In any Delta ABC, prove that the area Delta = (b^(2) + c^(2) - a^(2))/(4 cot A )

The combined equation of three sides of a triangle is (x^2-y^2)(2x+3y-6)=0 . If (-2,a) is an interior point and (b ,1) is an exterior point of the triangle, then (a) 2 < a < (10)/3 (b) -2 < a < (10)/3 (c) -1< b < 9/2 (d) -1< b < 1

Cosine formula: With usual notations in any triangle ABC Prove that : cos A= (b^(2)+c^(2)-a^(2))/(2bc) .

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

Let A be any 3xx2 matrix. Then prove that det. (A A^(T))=0 .