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.

A kite is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a circle tangent to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals.^{ [1] }

A rhombus is an orthodiagonal quadrilateral with two pairs of parallel sides (that is, an orthodiagonal quadrilateral that is also a parallelogram).

A square is a limiting case of both a kite and a rhombus.

Orthodiagonal equidiagonal quadrilaterals 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. An orthodiagonal quadrilateral that is also equidiagonal is a midsquare quadrilateral because its Varignon parallelogram is a square. Its area can be expressed purely in terms of its sides.

For any orthodiagonal quadrilateral, the sum of the squares of two opposite sides equals that of the other two opposite sides: for successive sides *a*, *b*, *c*, and *d*, we have ^{ [2] }^{ [3] }

This follows from the Pythagorean theorem, by which either of these two sums of two squares can be expanded to equal the sum of the four squared distances from the quadrilateral's vertices to the point where the diagonals intersect. Conversely, any quadrilateral in which *a*^{2} + *c*^{2} = *b*^{2} + *d*^{2} must be orthodiagonal.^{ [4] } This can be proved in a number of ways, including using the law of cosines, vectors, an indirect proof, and complex numbers.^{ [5] }

The diagonals of a convex quadrilateral are perpendicular if and only if the two bimedians have equal length.^{ [5] }

According to another characterization, the diagonals of a convex quadrilateral *ABCD* are perpendicular if and only if

where *P* is the point of intersection of the diagonals. From this equation it follows almost immediately that the diagonals of a convex quadrilateral are perpendicular if and only if the projections of the diagonal intersection onto the sides of the quadrilateral are the vertices of a cyclic quadrilateral.^{ [5] }

A convex quadrilateral is orthodiagonal if and only if its Varignon parallelogram (whose vertices are the midpoints of its sides) is a rectangle.^{ [5] } A related characterization states that a convex quadrilateral is orthodiagonal if and only if the midpoints of the sides and the feet of the four maltitudes are eight concyclic points; the **eight point circle**. The center of this circle is the centroid of the quadrilateral. The quadrilateral formed by the feet of the maltitudes is called the *principal orthic quadrilateral*.^{ [6] }

If the normals to the sides of a convex quadrilateral *ABCD* through the diagonal intersection intersect the opposite sides in *R*, *S*, *T*, *U*, and *K*, *L*, *M*, *N* are the feet of these normals, then *ABCD* is orthodiagonal if and only if the eight points *K*, *L*, *M*, *N*, *R*, *S*, *T* and *U* are concyclic; the *second eight point circle*. A related characterization states that a convex quadrilateral is orthodiagonal if and only if *RSTU* is a rectangle whose sides are parallel to the diagonals of *ABCD*.^{ [5] }

There are several metric characterizations regarding the four triangles formed by the diagonal intersection *P* and the vertices of a convex quadrilateral *ABCD*. Denote by *m*_{1}, *m*_{2}, *m*_{3}, *m*_{4} the medians in triangles *ABP*, *BCP*, *CDP*, *DAP* from *P* to the sides *AB*, *BC*, *CD*, *DA* respectively. If *R*_{1}, *R*_{2}, *R*_{3}, *R*_{4} and *h*_{1}, *h*_{2}, *h*_{3}, *h*_{4} denote the radii of the circumcircles and the altitudes respectively of these triangles, then the quadrilateral *ABCD* is orthodiagonal if and only if any one of the following equalities holds:^{ [5] }

Furthermore, a quadrilateral *ABCD* with intersection *P* of the diagonals is orthodiagonal if and only if the circumcenters of the triangles *ABP*, *BCP*, *CDP* and *DAP* are the midpoints of the sides of the quadrilateral.^{ [5] }

A few metric characterizations of tangential quadrilaterals and orthodiagonal quadrilaterals are very similar in appearance, as can be seen in this table.^{ [5] } The notations on the sides *a*, *b*, *c*, *d*, the circumradii *R*_{1}, *R*_{2}, *R*_{3}, *R*_{4}, and the altitudes *h*_{1}, *h*_{2}, *h*_{3}, *h*_{4} are the same as above in both types of quadrilaterals.

Tangential quadrilateral | Orthodiagonal quadrilateral |
---|---|

The area *K* of an orthodiagonal quadrilateral equals one half the product of the lengths of the diagonals *p* and *q*:^{ [7] }

Conversely, any convex quadrilateral where the area can be calculated with this formula must be orthodiagonal.^{ [5] } The orthodiagonal quadrilateral has the biggest area of all convex quadrilaterals with given diagonals.

- Orthodiagonal quadrilaterals are the only quadrilaterals for which the sides and the angle formed by the diagonals do not uniquely determine the area.
^{ [3] }For example, two rhombi both having common side*a*(and, as for all rhombi, both having a right angle between the diagonals), but one having a smaller acute angle than the other, have different areas (the area of the former approaching zero as the acute angle approaches zero). - If squares are erected outward on the sides of any quadrilateral (convex, concave, or crossed), then their centres (centroids) are the vertices of an orthodiagonal quadrilateral that is also equidiagonal (that is, having diagonals of equal length). This is called Van Aubel's theorem.
- Each side of an orthodiagonal quadrilateral has at least one common point with the Pascal points circle.
^{ [8] }

For a cyclic orthodiagonal quadrilateral (one that can be inscribed in a circle), suppose the intersection of the diagonals divides one diagonal into segments of lengths *p*_{1} and *p*_{2} and divides the other diagonal into segments of lengths *q*_{1} and *q*_{2}. Then^{ [9] } (the first equality is Proposition 11 in Archimedes Book of Lemmas)

where *D* is the diameter of the circumcircle. This holds because the diagonals are perpendicular chords of a circle. These equations yield the circumradius expression

or, in terms of the sides of the quadrilateral, as^{ [2] }

It also follows that^{ [2] }

Thus, according to Euler's quadrilateral theorem, the circumradius can be expressed in terms of the diagonals *p* and *q*, and the distance *x* between the midpoints of the diagonals as

A formula for the area *K* of a cyclic orthodiagonal quadrilateral in terms of the four sides is obtained directly when combining Ptolemy's theorem and the formula for the area of an orthodiagonal quadrilateral. The result is^{ [10] }^{:p.222}

- In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect.
^{ [2] } - Brahmagupta's theorem states that for a cyclic orthodiagonal quadrilateral, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side.
^{ [2] } - If an orthodiagonal quadrilateral is also cyclic, the distance from the circumcenter (the center of the circumscribed circle) to any side equals half the length of the opposite side.
^{ [2] } - In a cyclic orthodiagonal quadrilateral, the distance between the midpoints of the diagonals equals the distance between the circumcenter and the point where the diagonals intersect.
^{ [2] }

For every orthodiagonal quadrilateral, we can inscribe two infinite sets of rectangles:

- (i) a set of rectangles whose sides are parallel to the diagonals of the quadrilateral
- (ii) a set of rectangles defined by Pascal-points circles.
^{ [11] }

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 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 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 geometry, a set of points are said to be **concyclic** if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, but four or more such points in the plane are not necessarily concyclic.

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 geometry, given a triangle *ABC* and a point *P* on its circumcircle, the three closest points to *P* on lines *AB*, *AC*, and *BC* are collinear. The line through these points is the **Simson line** of *P*, named for Robert Simson. The concept was first published, however, by William Wallace in 1799.

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 **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, 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.

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.

In geometry, the **Newton–Gauss line** is the line joining the midpoints of the three diagonals of a complete quadrangle.

- ↑ Josefsson, Martin (2010), "Calculations concerning the tangent lengths and tangency chords of a tangential quadrilateral" (PDF),
*Forum Geometricorum*,**10**: 119–130. - 1 2 3 4 5 6 7 Altshiller-Court, N. (2007),
*College Geometry*, Dover Publications. Republication of second edition, 1952, Barnes & Noble, pp. 136-138. - 1 2 Mitchell, Douglas, W. (2009), "The area of a quadrilateral",
*The Mathematical Gazette*,**93**(July): 306–309. - ↑ Ismailescu, Dan; Vojdany, Adam (2009), "Class preserving dissections of convex quadrilaterals" (PDF),
*Forum Geometricorum*,**9**: 195–211. - 1 2 3 4 5 6 7 8 9 Josefsson, Martin (2012), "Characterizations of Orthodiagonal Quadrilaterals" (PDF),
*Forum Geometricorum*,**12**: 13–25. - ↑ Mammana, Maria Flavia; Micale, Biagio; Pennisi, Mario (2011), "The Droz-Farny Circles of a Convex Quadrilateral" (PDF),
*Forum Geometricorum*,**11**: 109–119. - ↑ Harries, J. (2002), "Area of a quadrilateral",
*The Mathematical Gazette*,**86**(July): 310–311 - ↑ David, Fraivert (2017), "Properties of a Pascal points circle in a quadrilateral with perpendicular diagonals" (PDF),
*Forum Geometricorum*,**17**: 509–526. - ↑ Posamentier, Alfred S.; Salkind, Charles T. (1996),
*Challenging Problems in Geometry*(second ed.), Dover Publications, pp. 104–105, #4–23. - ↑ Josefsson, Martin (2016), "Properties of Pythagorean quadrilaterals",
*The Mathematical Gazette*,**100**(July): 213–224. - ↑ David, Fraivert (2019), "A Set of Rectangles Inscribed in an Orthodiagonal Quadrilateral and Defined by Pascal-Points Circles",
*Journal for Geometry and Graphics*,**23**: 5–27.

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