The inradiuses of the triangles that are generated from triangulations of these shapes will always sum to a constant value according to the Japanese Theorem.
The inradiuses of the triangles that are generated from triangulations of these shapes will always sum to a constant value according to the Japanese Theorem. The side lengths of these shapes are the inputs to Parameshvara's formula. The product of the diagonals of these shapes will equal the sum of the products of the opposite sides according to Ptolemy's theorem. Integer lengths for the sides, diagonals, and area characterize the (*) "Brahmagupta" type of these shapes. Euclid's 22nd proposition in the third book of his Elements states that quadrilaterals will have this property if and only if the opposite angles are supplementary. A circumcenter is well-defined on, for 10 points, what quadrilaterals, which have vertices that all lie on a single circle?
The inradiuses of the triangles that are generated from triangulations of these shapes will always sum to a constant value according to the Japanese Theorem.
cyclic quadrilateral [accept inscribed quadrilateral; prompt on quadrilateral]