A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle. The opposite angles of a cyclic quadrilateral are supplementary, which means that their sum is equal to 180 degrees.

The cyclic quadrilateral formula is used to find the length of one side of a cyclic quadrilateral when the lengths of the other three sides and two opposite angles are known. The formula is given as:

a² = b² + c² – 2bc cos(A) where:

• a is the length of the unknown side
• b and c are the lengths of the other two sides
• A is the opposite angle to the side a

Alternatively, if the lengths of all four sides and one angle are known, the cyclic quadrilateral formula can be used to find the value of the remaining three angles. The formula is given as:

cos(A) = (b² + d² – a² – c²) / 2bd cos(B) = (a² + c² – b² – d²) / 2ac cos(C) = (b² + d² – a² – c²) / 2bd cos(D) = (a² + c² – b² – d²) / 2ac

where:

• A, B, C, and D are the four angles of the cyclic quadrilateral
• a, b, c, and d are the lengths of the four sides

Example

here’s an example of how to use the cyclic quadrilateral formula:

Suppose we have a cyclic quadrilateral with sides of length 6, 7, and 8, and the opposite angle to the side of length 8 is 120 degrees. We want to find the length of the remaining side.

Using the cyclic quadrilateral formula, we have: a² = b² + c² – 2bc cos(A) a² = 6² + 7² – 2(6)(7)cos(120) a² = 36 + 49 + 42 a² = 127 a ≈ 11.27

So the length of the remaining side is approximately 11.27.