The sea dragon formula for the area of a circle inside a quadrilateral is S = √((p – a)(p – b)(p – c)(p – d)) to the square root, where a, b, c, d are the four sides of the quadrilateral and p is the half circumference (i.e. half the circumference of the quadrilateral), which can be used to calculate the area of a circle inside a quadrilateral of any shape.

 here is a recapitulation of the derivation process of Brahmagupta’s formula:

1. Setting up the Quadrilateral and Variables:
   Consider a quadrilateral ABCD with sides a, b, c, and d. Assume that the diagonals AC and BD are perpendicular to each other.

2. Utilizing the Pythagorean Theorem:
   Since AC and BD are perpendicular, the quadrilateral ABCD can be divided into two right triangles, ΔABC and ΔADC.
   Let AC = x and BD = y. By the Pythagorean theorem, we have:
   x^2 = a^2 + b^2
   y^2 = c^2 + d^2

3. Calculating the Area of the Quadrilateral:
   The area of the quadrilateral ABCD, denoted as S, can be expressed as the sum of the areas of the two right triangles:
   S = 1/2 × x × y
   Substituting the expressions for x^2 and y^2 into the above equation, we get:
   S = 1/2 × √((a^2 + b^2)(c^2 + d^2))

4. Expressing Side Lengths in Terms of Semiperimeter:
   The semiperimeter p is defined as p = (a + b + c + d)/2. It can be rearranged to express each side length in terms of the others and p:
   a = 2p – (b + c + d)
   b = 2p – (a + c + d)
   c = 2p – (a + b + d)
   d = 2p – (a + b + c)

5. Deriving Brahmagupta’s Formula:
   Substituting the expressions for a, b, c, and d into the area formula and performing a series of algebraic manipulations and simplifications, we arrive at the standard form of Brahmagupta’s formula:
   S = √((p – a)(p – b)(p – c)(p – d))

This derivation relies on the Pythagorean theorem, the calculation of quadrilateral area, and algebraic manipulations. It is important to note that this derivation assumes that the diagonals of the quadrilateral are perpendicular, thus making it applicable to quadrilaterals with perpendicular diagonals, including circumscribed quadrilaterals. If the diagonals of the quadrilateral are not perpendicular, then this formula is not applicable.