![Orthodiagonal quadrilateral](https://upload.wikimedia.org/wikipedia/commons/1/16/Orthodiagonal_quadrilateral.svg)
In
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 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.
Special cases
A
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
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
. 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 con ...
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
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. More ...
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 of ...
sides (that is, an orthodiagonal quadrilateral that is also a
parallelogram).
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 ope ...
for their diameter among all quadrilaterals, solving the ''n'' = 4 case of the
biggest little polygon
In geometry, the biggest little polygon for a number ''n'' is the ''n''-sided polygon that has diameter one (that is, every two of its points are within unit distance of each other) and that has the largest area among all diameter-one ''n''-gons. ...
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
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 wa ...
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, .]
:
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.
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 known ...
, 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 b ...
the two
bimedians have equal length.
[
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
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 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 wa ...
(whose vertices are the midpoint
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.
Formula
The midpoint of a segment in ''n''-dimen ...
s of its sides) is a rectangle.[ 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
In geometry, a set of points are said to be concyclic (or cocyclic) 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 ar ...
; 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''.
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 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
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
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:[
*
*
*
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 the ...
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'':
:
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 a ...
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 eccentri ...
s (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 ...
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.
*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 s ...
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' Book of Lemmas
The ''Book of Lemmas'' or ''Book of Assumptions'' (Arabic ''Maʾkhūdhāt Mansūba ilā Arshimīdis'') is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositio ...
)
:
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 fo ...
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[
:
It also follows that][
:
Thus, according to ]Euler's quadrilateral theorem
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 ...
, 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
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 mathematician ...
and the formula for the area of an orthodiagonal quadrilateral. The result is[.]
:
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 ]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 ...
(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
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