Pitot theorem

In every tangential quadrilateral, the sums of the lengths of each pair of opposite sides are equal. The theorem gives a necessary condition for a quadrilateral to have an incircle.

Tangential quadrilateral: A convex quadrilateral whose four sides are tangent to one circle. The theorem applies to exactly this class of quadrilaterals.
Quadrilateral: A polygon with four sides and four vertices. The theorem compares the two pairs of opposite sides of this polygon.
Heron's formula: A formula that gives a triangle’s area from its three side lengths. The analogous tangential-polygon area formula uses the semiperimeter identified by Pitot’s equality.
Mathematical proof: A deductive argument establishing a mathematical statement from accepted premises. The theorem follows from a short proof using equal tangent segments.
Incircle: A circle tangent to every side of a polygon. Its contact points divide the sides into equal tangent segments at each vertex.
Convex polygon: A polygon whose interior contains every line segment joining two of its points. The standard tangential-quadrilateral setting is convex.
Tangential polygon: A polygon whose sides are all tangent to one circle. Pitot’s theorem is the four-sided case of perimeter relations for tangential polygons.
Euclidean geometry: The study of geometric figures and relations in a flat plane. Pitot’s theorem is a classical result within plane geometry.
Tangent-segment theorem: Two tangent segments drawn from the same external point to a circle have equal lengths. Equal tangent lengths at each vertex produce the opposite-side sum equality.
Opposite sides: Sides of a polygon that do not share a vertex. The theorem equates the sums formed by the two opposite-side pairs.

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Japanese theoremCompared with: It is another quadrilateral equality involving opposite pairs, but concerns side lengths and tangency.