Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, an orthodiagonal quadrilateral is a

quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...

in which the

diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...

s cross at

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. Th ...

s. In other words, it is a four-sided figure in which the

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...

s between non-adjacent vertices are

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

(perpendicular) to each other.

Special cases

kite A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the fac ...

is an orthodiagonal quadrilateral in which one diagonal is a line of symmetry. The kites are exactly the orthodiagonal quadrilaterals that contain a

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

to all four of their sides; that is, the kites are the tangential orthodiagonal quadrilaterals. A

rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) 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 ...

is an orthodiagonal quadrilateral with two pairs of

parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...

sides (that is, an orthodiagonal quadrilateral that is also a

parallelogram 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 eq ...

). A

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

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 Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...

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 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 ...

because its

Varignon parallelogram 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 ...

is a square. Its area can be expressed purely in terms of its sides.

Characterizations

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 . Republication of second edition, 1952, Barnes & Noble, pp. 136-138.Mitchell, . :

\displaystyle a^2+c^2=b^2+d^2.

This follows from the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...

, 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 In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the categorical proposit ...

, any quadrilateral in which ''a''² + ''c''² = ''b''² + ''d''² must be orthodiagonal. This can be proved in a number of ways, including using the

law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...

, vectors, an

indirect proof In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also know ...

, and

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

s.. The diagonals of a

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

quadrilateral are perpendicular

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...

the two bimedians have equal length. According to another characterization, the diagonals of a convex quadrilateral ''ABCD'' are perpendicular if and only if :

\angle PAB + \angle PBA + \angle PCD + \angle PDC = \pi

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 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 ''c ...

A convex quadrilateral is orthodiagonal if and only if its

(whose vertices are the midpoints of its sides) is a

rectangle 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 (360°/4 = 90°); or a parallelogram contain ...

. 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 In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...

of the quadrilateral. The quadrilateral formed by the feet of the maltitudes is called the ''principal orthic quadrilateral''. Second 8 point circle

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''. There are several metric characterizations regarding the four

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

s 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 The Medes (Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, the ...

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

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

s and the

altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

s respectively of these triangles, then the quadrilateral ''ABCD'' is orthodiagonal if and only if any one of the following equalities holds: *

m_1^2+m_3^2=m_2^2+m_4^2

R_1^2+R_3^2=R_2^2+R_4^2

\frac+\frac=\frac+\frac

Furthermore, a quadrilateral ''ABCD'' with intersection ''P'' of the diagonals is orthodiagonal if and only if the

circumcenter In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

s of the triangles ''ABP'', ''BCP'', ''CDP'' and ''DAP'' are the midpoints of the sides of the quadrilateral.

Comparison with a tangential quadrilateral

A few metric characterizations of

tangential quadrilateral In Euclidean geometry, a tangential quadrilateral (sometimes just tangent 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 ...

s and orthodiagonal quadrilaterals are very similar in appearance, as can be seen in this table. 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.

Area

The area ''K'' of an orthodiagonal quadrilateral equals one half the product of the lengths of the diagonals ''p'' and ''q'': :

K = \frac.

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

Other properties

*Orthodiagonal quadrilaterals are the only quadrilaterals for which the sides and the

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...

formed by the diagonals do not uniquely determine the area. 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 In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ar ...

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

centre Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentricity ...

s (

s) 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 In plane geometry, Van Aubel's theorem describes a relationship between squares constructed on the sides of a quadrilateral. Starting with a given convex quadrilateral, construct a square, external to the quadrilateral, on each side. Van Aubel's ...

. *Each side of an orthodiagonal quadrilateral has at least one common point with the Pascal points circle..

Properties of orthodiagonal quadrilaterals that are also cyclic

Circumradius and area

For a

cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in so ...

orthodiagonal quadrilateral (one that can be

inscribed {{unreferenced, date=August 2012 An inscribed triangle of a circle In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figu ...

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. (the first equality is Proposition 11 in

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientis ...

' Book of Lemmas) :

D^2=p_1^2+p_2^2+q_1^2+q_2^2=a^2+c^2=b^2+d^2

where ''D'' is the

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...

of the circumcircle. This holds because the diagonals are perpendicular chords of a circle. These equations yield the circumradius expression :

R = \tfrac\sqrt

or, in terms of the sides of the quadrilateral, as :

R = \tfrac\sqrt=\tfrac\sqrt.

It also follows that :

a^2+b^2+c^2+d^2=8R^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 :

R = \sqrt\,.

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 In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematici ...

and the formula for the area of an orthodiagonal quadrilateral. The result is. :

K=\tfrac(ac+bd).

Other properties

*In a cyclic orthodiagonal quadrilateral, the anticenter coincides with the point where the diagonals intersect. *

Brahmagupta's theorem In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. I ...

states that for a cyclic orthodiagonal quadrilateral, the perpendicular from any side through the point of intersection of the diagonals bisects the opposite side. *If an orthodiagonal quadrilateral is also cyclic, the distance from the

(the center of the circumscribed circle) to any side equals half the length of the opposite side. *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.

Infinite sets of inscribed rectangles

A set of inscribed rectangles whose sides are parallel to the diagonals of the quadrilateral

A set of inscribed rectangles defined by Pascal-points circles

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..

References

{{Polygons Types of quadrilaterals