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If the diagonals of a cyclic quadrilateral are diameters of the circle through the opposite vertices of the quadrilateral, prove that the quadrilateral is

a rectangle


ABCD is a cyclic quadrilateral and AC and BD are the diameters of the circle.
To prove: ABCD is a rectangle.
Proof: To prove that ABCD is a rectangle, we only need to prove that its all angles are 90 degrees.

In semi-ciecle ADC,
angle ADC = 90 degrees (since, the line segment from the two ends of the diameter of a semi-circle makes 90 degrees on the circumference.)
Similarly, we can prove that angle ABC, angle BCD and angle DAB is equal to 90 degrees.

Therefore, ABCD is a rectangle.
Hence, Proved.
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