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Question
If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
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Solution
Let ABCD be a cyclic quadrilateral having diagonals BD and AC, intersecting each other at point O.
∠BAD = `1/2angleBOD`
= `180^@/2`
= 90° ...(Consider BD as a chord)
∠BCD + ∠BAD = 180° ...(Cyclic quadrilateral)
∠BCD = 180° − 90° = 90°
∠ADC = `1/2angleAOC`
= `1/2(180^@)`
= 90° ...(Considering AC as a chord)
∠ADC + ∠ABC = 180° ...(Cyclic quadrilateral)
90° + ∠ABC = 180°
∠ABC = 90°
Each interior angle of a cyclic quadrilateral is of 90°. Hence, it is a rectangle.
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