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Question

ABCD is a rectangle. Prove that the centre of the circle through A, B, C, D is the point of intersection of its diagonals.


Solution

ANSWER:
Given: ABCD is a cyclic rectangle whose diagonals intersect at O.
To prove: O is the centre of the circle.
Proof:
Here, ∠BCD = 90° [Since it is a rectangle]
So, BD is the diameter of the circle (if the angle made by the chord at the circle is right angle, then the chord is
the diameter).
Also, diagonals of a rectangle bisect each other and are equal.
∴ OA = OB = OC = OD
BD is the diameter.∴ BO and OD are the radius.
Thus, O is the centre of the circle.
Also, the centre of the circle is circumscribing the cyclic rectangle.
Hence, O is the point of intersection of the diagonals of ABCD.

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