For a cyclic quadrilateral with given sides a, b, c, and d, the formula for the area is given by

Where s = (a + b + c + d)/2 known as the semi-perimeter.

**Derivation of Formula**

Cosine law for triangle ABC

Cosine law for triangle ADC

note that cos (180° - x) = -cos x

Equate the two x^{2}

Area of ABCD

note that sin (180° - x) = sin x

Square both sides of the equation

From

Note that a + b + c + d = P, the perimeter. Thus,

Substitute 1 - cos^{2} θ to the equation of A^{2} above.

Recall that , the semi-perimeter. Thus,

Finally,

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