Japanese theorem

In a cyclic quadrilateral, the sums of the inradii of the two triangles on either pair of opposite sides are equal. The triangles are formed by the quadrilateral’s diagonals.

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Inradius: The radius of a circle inscribed in a polygon, tangent to each of its sides. The theorem compares the inradii of four triangles created by the diagonals.
Triangle: A polygon with three sides and three interior angles. Each region formed by the diagonals is a triangle whose inradius enters the theorem.
Japanese sangaku: Wooden votive tablets bearing mathematical problems and diagrams, offered at Japanese shrines and temples. The theorem is associated with the geometry tradition preserved on these tablets.
Orthodiagonal quadrilateral: A quadrilateral whose diagonals intersect at a right angle. Perpendicular diagonals give a particularly symmetric setting for testing the inradius sums.
Heron’s formula: A formula giving a triangle’s area from its three side lengths. Combining area with semiperimeter expresses each triangle’s inradius using its sides.
Incircle: The largest circle contained in a polygon and tangent to every side. Its radius is the inradius being summed in the theorem.
Sangaku geometry: A body of Japanese geometry problems recorded on votive tablets during the Edo period and later. Its circle and polygon problems provide the historical setting for the theorem’s name.
Rectangle: A quadrilateral with four right angles. Its equal opposite sides and symmetric diagonals make the paired triangles easy to compare.
Semiperimeter: Half the perimeter of a geometric figure. Triangle inradius equals area divided by semiperimeter, the key relation behind the equality.
Circle: The set of points in a plane at a fixed distance from a center. The quadrilateral’s vertices lie on a circle, which imposes its angle constraints.