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.^{ [1] }

Examples of equidiagonal quadrilaterals include the isosceles trapezoids, rectangles and squares.

Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles π/3, 5π/12, 5π/6, and 5π/12.^{ [2] }

A convex quadrilateral is equidiagonal if and only if its Varignon parallelogram, the parallelogram formed by the midpoints of its sides, is a rhombus. An equivalent condition is that the bimedians of the quadrilateral (the diagonals of the Varignon parallelogram) are perpendicular.^{ [3] }

A convex quadrilateral with diagonal lengths and and bimedian lengths and is equidiagonal if and only if^{ [4] }^{:Prop.1}

The area *K* of an equidiagonal quadrilateral can easily be calculated if the length of the bimedians *m* and *n* are known. A quadrilateral is equidiagonal if and only if^{ [5] }^{:p.19;}^{ [4] }^{:Cor.4}

This is a direct consequence of the fact that the area of a convex quadrilateral is twice the area of its Varignon parallelogram and that the diagonals in this parallelogram are the bimedians of the quadrilateral. Using the formulas for the lengths of the bimedians, the area can also be expressed in terms of the sides *a, b, c, d* of the equidiagonal quadrilateral and the distance *x* between the midpoints of the diagonals as^{ [5] }^{:p.19}

Other area formulas may be obtained from setting *p* = *q* in the formulas for the area of a convex quadrilateral.

A parallelogram is equidiagonal if and only if it is a rectangle,^{ [6] } and a trapezoid is equidiagonal if and only if it is an isosceles trapezoid. The cyclic equidiagonal quadrilaterals are exactly the isosceles trapezoids.

There is a duality between equidiagonal quadrilaterals and orthodiagonal quadrilaterals: a quadrilateral is equidiagonal if and only if its Varignon parallelogram is orthodiagonal (a rhombus), and the quadrilateral is orthodiagonal if and only if its Varignon parallelogram is equidiagonal (a rectangle).^{ [3] } Equivalently, a quadrilateral has equal diagonals if and only if it has perpendicular bimedians, and it has perpendicular diagonals if and only if it has equal bimedians.^{ [7] } Silvester (2006) gives further connections between equidiagonal and orthodiagonal quadrilaterals, via a generalization of van Aubel's theorem.^{ [8] }

Quadrilaterals that are both orthodiagonal and equidiagonal, and in which the diagonals are at least as long as all of the quadrilateral's sides, have the maximum area for their diameter among all quadrilaterals, solving the *n* = 4 case of the biggest little polygon problem. The square is one such quadrilateral, but there are infinitely many others. Equidiagonal, orthodiagonal quadrilaterals have been referred to as *midsquare quadrilaterals*^{ [4] }^{:p. 137} because they are the only ones for which the Varignon parallelogram (with vertices at the midpoints of the quadrilateral's sides) is a square. Such a quadrilateral, with successive sides *a, b, c, d*, has area^{ [4] }^{:Thm.16}

A midsquare parallelogram is exactly a square.

- example of a midsquare quadrilateral
- a midsquare trapezoid
- a midsquare kite

In Euclidean plane geometry, a **quadrilateral** is a polygon with four edges (sides) and four vertices (corners). Other names for quadrilateral include **quadrangle**, **tetragon**, and **4-gon**. A quadrilateral with vertices , , and is sometimes denoted as .

In Euclidean plane geometry, a **rectangle** is 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. It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term **oblong** is occasionally used to refer to a non-square rectangle. A rectangle with vertices *ABCD* would be denoted as *ABCD*.

In elementary geometry, the property of being **perpendicular** (**perpendicularity**) is the relationship between two lines which meet at a right angle. The property extends to other related geometric objects.

In Euclidean geometry, a **kite ** is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other instead of being adjacent. Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape and which are in turn named for a bird. Kites are also known as **deltoids**, but the word "deltoid" may also refer to a deltoid curve, an unrelated geometric object.

In Euclidean geometry, a **parallelogram** is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

In geometry, **bisection** is the division of something into two equal or congruent parts, usually by a line, which is then called a *bisector*. The most often considered types of bisectors are the *segment bisector* and the *angle bisector*.

In plane Euclidean geometry, a **rhombus** is a quadrilateral whose four sides all have the same length. Another name is **equilateral quadrilateral**, since equilateral means that all of its sides are equal in length. The rhombus is often called a **diamond**, after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a **lozenge**, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

In Euclidean geometry, a **cyclic quadrilateral** or **inscribed quadrilateral** is a quadrilateral whose vertices all lie on a single circle. This circle is called the *circumcircle* or circumscribed circle, and the vertices are said to be *concyclic*. The center of the circle and its radius are called the *circumcenter* and the *circumradius* respectively. Other names for these quadrilaterals are **concyclic quadrilateral** and **chordal quadrilateral**, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a **trapezium** in English outside North America, but as a **trapezoid** in American and Canadian English. The parallel sides are called the *bases* of the trapezoid and the other two sides are called the *legs* or the lateral sides. A *scalene trapezoid* is a trapezoid with no sides of equal measure, in contrast to the special cases below.

In geometry, the **midpoint** is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment.

In Euclidean geometry, an **isosceles trapezoid** is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of the same measure. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides are parallel, and the two other sides are of equal length. The diagonals are also of equal length. The base angles of an isosceles trapezoid are equal in measure.

In geometry, a **square** is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices *ABCD* would be denoted *ABCD*.

In Euclidean geometry, a **tangential quadrilateral** or **circumscribed quadrilateral** is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the *incenter* and its radius is called the *inradius*. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called *circumscribable quadrilaterals*, *circumscribing quadrilaterals*, and *circumscriptible quadrilaterals*. Tangential quadrilaterals are a special case of tangential polygons.

**Varignon's theorem** is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the **Varignon parallelogram**, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof was published posthumously in 1731.

In Euclidean geometry, a **bicentric quadrilateral** is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called *inradius* and *circumradius*, and *incenter* and *circumcenter* respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are **chord-tangent quadrilateral** and **inscribed and circumscribed quadrilateral**. It has also rarely been called a *double circle quadrilateral* and *double scribed quadrilateral*.

In Euclidean geometry, an **orthodiagonal quadrilateral** is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

In Euclidean geometry, an **ex-tangential quadrilateral** is a convex quadrilateral where the *extensions* of all four sides are tangent to a circle outside the quadrilateral. It has also been called an **exscriptible quadrilateral**. The circle is called its *excircle*, its radius the *exradius* and its center the *excenter*. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

In Euclidean geometry, a **tangential trapezoid**, also called a **circumscribed trapezoid**, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or *inscribed circle*. It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the parallel sides are called the bases and the other two sides the *legs*. The legs can be equal, but they don't have to be.

In Euclidean geometry, a **right kite** is a kite that can be inscribed in a circle. That is, it is a kite with a circumcircle. Thus the right kite is a convex quadrilateral and has two opposite right angles. If there are exactly two right angles, each must be between sides of different lengths. All right kites are bicentric quadrilaterals, since all kites have an incircle. One of the diagonals divides the right kite into two right triangles and is also a diameter of the circumcircle.

**Euler's quadrilateral theorem** or **Euler's law on quadrilaterals**, named after Leonhard Euler (1707–1783), describes a relation between the sides of a convex quadrilateral and its diagonals. It is a generalisation of the parallelogram law which in turn can be seen as generalisation of the Pythagorean theorem. Because of the latter the restatement of the Pythagorean theorem in terms of quadrilaterals is occasionally called the **Euler–Pythagoras theorem**.

- ↑ Colebrooke, Henry-Thomas (1817),
*Algebra, with arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bhascara*, John Murray, p. 58. - ↑ Ball, D.G. (1973), "A generalisation of π",
*Mathematical Gazette*,**57**(402): 298–303, doi:10.2307/3616052 , Griffiths, David; Culpin, David (1975), "Pi-optimal polygons",*Mathematical Gazette*,**59**(409): 165–175, doi:10.2307/3617699 . - 1 2 de Villiers, Michael (2009),
*Some Adventures in Euclidean Geometry*, Dynamic Mathematics Learning, p. 58, ISBN 9780557102952 . - 1 2 3 4 Josefsson, Martin (2014), "Properties of equidiagonal quadrilaterals",
*Forum Geometricorum*,**14**: 129–144. - 1 2 Josefsson, Martin (2013), "Five Proofs of an Area Characterization of Rectangles" (PDF),
*Forum Geometricorum*,**13**: 17–21. - ↑ Gerdes, Paulus (1988), "On culture, geometrical thinking and mathematics education",
*Educational Studies in Mathematics*,**19**(2): 137–162, doi:10.1007/bf00751229, JSTOR 3482571 . - ↑ Josefsson, Martin (2012), "Characterizations of Orthodiagonal Quadrilaterals" (PDF),
*Forum Geometricorum*,**12**: 13–25. See in particular Theorem 7 on p. 19. - ↑ Silvester, John R. (2006), "Extensions of a theorem of Van Aubel",
*The Mathematical Gazette*,**90**(517): 2–12, JSTOR 3621406 .

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