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Prove that any rectangle can be inscribed in a circle.

Last updated date: 13th Jul 2024
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Hint: Prove that any rectangle is a cyclic quadrilateral.
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To prove the above statement, assume $\square $ ABCD to be any rectangle.
According to the properties of Rectangle, we can write,
\[\angle A = \angle B = \angle C = \angle D = 90^\circ \] ………………….. (1)
(All angles of a rectangle are always\[90^\circ \])
As we have to prove that rectangle can be inscribed in a circle which means any rectangle is a
Cyclic Quadrilateral, we should know the necessary and sufficient condition for a quadrilateral
to be Cyclic which is given below,
Condition: Any quadrilateral can be a cyclic quadrilateral if it’s opposite angles is
supplementary i.e. their summation should be equal to $180^\circ $
From figure and equation (1) we can write,
\[\angle A=\angle C=90{}^\circ \] $\therefore \angle A+\angle C=90{}^\circ +90{}^\circ =180{}^\circ
$…………………………. (2)
\[\angle B = \angle D = 90^\circ \] $\therefore \angle B + \angle D = 90^\circ + 90^\circ =
180^\circ $……………………………. (3)

From (2) and (3) it is clear that the pairs of opposite angles of rectangle ABCD are Supplementary.
And therefore, Rectangle ABCD is a Cyclic Quadrilateral.
As we assumed rectangle ABCD to be any rectangle, therefore we can generalize our statement as,
Any rectangle can be a cyclic quadrilateral and therefore any rectangle can be inscribed in a circle.
Hence proved.

Note:I have inscribed the above rectangle in a circle geometrically therefore it is proved experimentally also.
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If you are not getting it, think of the other simpler concept i.e. Semicircular angles are always right angles and therefore any right angle can be inscribed in a semicircle and we can easily consider two right angles in a rectangle to complete a circle.