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A root-phi rectangle divides into a pair of Kepler triangles (right triangles with edge lengths in geometric progression). The root-φ rectangle is a dynamic rectangle but not a root rectangle. Its diagonal equals φ times the length of the shorter side. If a root-φ rectangle is divided by a diagonal, the result is two congruent Kepler triangles.
A rhombus has an inscribed circle, while a rectangle has a circumcircle. A rhombus has an axis of symmetry through each pair of opposite vertex angles, while a rectangle has an axis of symmetry through each pair of opposite sides. The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length.
For the general quadrilateral (with four sides not necessarily equal) Euler's quadrilateral theorem states + + + = + +, where is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that x = 0 {\displaystyle x=0} for a parallelogram, and so the general formula simplifies to the parallelogram law.
The central angle of a square is equal to 90° (360°/4). The external angle of a square is equal to 90°. The diagonals of a square are equal and bisect each other, meeting at 90°. The diagonal of a square bisects its internal angle, forming adjacent angles of 45°. All four sides of a square are equal. Opposite sides of a square are parallel.
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d 2, the right hand side of Ptolemy's relation is the sum a 2 + b 2.
In Euclidean geometry, an equidiagonal quadrilateral is a convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics , where quadrilaterals were classified first according to whether they were equidiagonal and then into more specialized types.
Two pairs of opposite angles are equal in measure. The diagonals bisect each other. One pair of opposite sides is parallel and equal in length. Adjacent angles are supplementary. Each diagonal divides the quadrilateral into two congruent triangles. The sum of the squares of the sides equals the sum
The sum of the perpendiculars from a regular n-gon's vertices to any line tangent to the circumcircle equals n times the circumradius. [4]: p. 73 The sum of the squared distances from the vertices of a regular n-gon to any point on its circumcircle equals 2nR 2 where R is the circumradius. [4]: p. 73