Ptolemy's inequality

For any quadrilateral, the product of its diagonals is at most the sum of the products of its opposite sides. Equality holds for cyclic quadrilaterals.

Ptolemy's theorem: For a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides. This is the equality case of the inequality, characterizing when its bound is sharp.
Quadrilateral: A polygon with four sides and four vertices. The inequality compares the four side lengths and two diagonals of this figure.
Four-point geometry: The study of geometric relations determined by four points and their pairwise distances. Ptolemy's inequality is a fundamental constraint on the distances among four planar points.
Euler's quadrilateral theorem: A relation between a quadrilateral's side lengths, diagonals, and the distance between its diagonal midpoints. It is another general quadrilateral identity, but tracks diagonal geometry rather than bounding their product.
Triangle inequality: In any triangle, the length of one side is no greater than the sum of the other two. A standard proof reduces the quadrilateral bound to triangle inequalities after a suitable construction.
Diagonal: A line segment joining two nonadjacent vertices of a polygon. The two diagonals form the left-hand product in the inequality.
Ptolemaic metric space: A metric space in which every four points satisfy Ptolemy's inequality. This definition extends the quadrilateral bound from the plane to abstract distance spaces.
Bretschneider's formula: A formula for a quadrilateral's area using its sides and opposite angles. It addresses quadrilateral geometry through area and angles instead of a product inequality.
Law of cosines: The law relating a triangle's side lengths to the cosine of one of its angles. Applying it to triangles formed by a diagonal gives an algebraic route to the bound.
Opposite sides: Sides of a polygon that do not share a vertex. Their two pairwise products make up the right-hand sum.

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