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In Euclidean plane geometry, a rectangle is a rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square.
If the quadrilateral is rectangle, then equation simplifies further since now the two diagonals are of equal length as well: 2 a 2 + 2 b 2 = 2 e 2 {\displaystyle 2a^{2}+2b^{2}=2e^{2}} Dividing by 2 yields the Euler–Pythagoras theorem:
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.
The ratio of a long diagonal to a side is 2, and the ratio of a short diagonal to a side is . A regular heptagon has 14 diagonals. The seven shorter ones equal each other, and the seven longer ones equal each other. The reciprocal of the side equals the sum of the reciprocals of a short and a long diagonal.
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.
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.
The diagonals of a rhombus intersect at equal angles, while the diagonals of a rectangle are equal in length. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle , and vice versa.
Equidiagonal quadrilateral: the diagonals are of equal length. Bisect-diagonal quadrilateral: one diagonal bisects the other into equal lengths. Every dart and kite is bisect-diagonal. When both diagonals bisect another, it's a parallelogram. Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.