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Facts : 1 1 Geometry 1.1 Brahmagupta s formula 1.2 Triangles 1.3 Brahmagupta s theorem 1.4 Pi 1.5 Measurements and constructions Geometry Brahmagupta s formula Diagram for reference Brahmagupta s most famous result in geometry is his formula for cyclic quadrilaterals
Facts : 2 Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure s area, 12.21
Facts : 3 The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral
Facts : 4 The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral
Facts : 5 Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case
Facts : 6 Heron s formula is a special case of this formula and it can be derived by setting one of the sides equal to zero
Facts : 7 Triangles Brahmagupta dedicated a substantial portion of his work to geometry
Facts : 8 One theorem gives the lengths of the two segments a triangle s base is divided into by its altitude: 12.22
Facts : 9 The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments
Facts : 10 The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment
Facts : 11 A triangle with rational sides a, b, c and rational area is of the form: for some rational numbers u, v, and w
Facts : 12 The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal
Facts : 13 The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes]
Facts : 14 He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral
Facts : 15 Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases
Facts : 16 Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals
Facts : 17 The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle]
Facts