Orthodiagonal quadrilateral

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An orthodiagonal quadrilateral (yellow). According to the characterization of these quadrilaterals, the two red squares on two opposite sides of the quadrilateral have the same total area as the two blue squares on the other pair of opposite sides. Orthodiagonal quadrilateral.svg
An orthodiagonal quadrilateral (yellow). According to the characterization of these quadrilaterals, the two red squares on two opposite sides of the quadrilateral have the same total area as the two blue squares on the other pair of opposite sides.

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.

Contents

Special cases

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.

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 [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 a2 + c2 = b2 + d2 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 m1, m2, m3, m4 the medians in triangles ABP, BCP, CDP, DAP from P to the sides AB, BC, CD, DA respectively. If R1, R2, R3, R4 and h1, h2, h3, h4 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]

Comparison with a tangential quadrilateral

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 R1, R2, R3, R4, and the altitudes h1, h2, h3, h4 are the same as above in both types of quadrilaterals.

Tangential quadrilateralOrthodiagonal quadrilateral

Area

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.

Other properties

Properties of orthodiagonal quadrilaterals that are also cyclic

Circumradius and area

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 p1 and p2 and divides the other diagonal into segments of lengths q1 and q2. 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

Other properties

Infinite sets of inscribed rectangles

A
B
C
D
{\displaystyle ABCD}
is an orthodiagonal quadrilateral,
P
1
X
1
Z
1
Y
1
{\displaystyle P_{1}X_{1}Z_{1}Y_{1}}
and
P
2
X
2
Z
2
Y
2
{\displaystyle P_{2}X_{2}Z_{2}Y_{2}}
are rectangles whose sides are parallel to the diagonals of the quadrilateral. A set of inscribed rectangles whose sides are parallel to the diagonals of the quadrilateral.png
is an orthodiagonal quadrilateral, and are rectangles whose sides are parallel to the diagonals of the quadrilateral.
A
B
C
D
{\displaystyle ABCD}
is an orthodiagonal quadrilateral.
P
1
{\displaystyle P_{1}}
and
Q
1
{\displaystyle Q_{1}}
are Pascal points formed by the circle
o
1
{\displaystyle \omega _{1}}
,
s
P
1
Q
1
{\displaystyle \sigma _{P_{1}Q_{1}}}
is Pascal-points circle which defines the rectangle
P
1
V
1
Q
1
W
1
{\displaystyle P_{1}V_{1}Q_{1}W_{1}}
.
P
2
{\displaystyle P_{2}}
and
Q
2
{\displaystyle Q_{2}}
are Pascal points formed by the circle
o
2
{\displaystyle \omega _{2}}
,
s
P
2
Q
2
{\displaystyle \sigma _{P_{2}Q_{2}}}
is Pascal-points circle which defines the rectangle
P
2
V
2
Q
2
W
2
{\displaystyle P_{2}V_{2}Q_{2}W_{2}}
. A set of inscribed rectangles defined by Pascal-points circles.png
is an orthodiagonal quadrilateral. and are Pascal points formed by the circle , is Pascal-points circle which defines the rectangle . and are Pascal points formed by the circle , is Pascal-points circle which defines the rectangle .

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]

Related Research Articles

Quadrilateral polygon with four sides and four corners

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 .

Rectangle Quadrilateral with four right angles

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.

Kite (geometry)

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.

Bisection

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.

Rhombus Quadrilateral in which all sides have the same length

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.

Cyclic quadrilateral Quadrilateral whose vertices can all fall on a single circle

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.

Trapezoid Convex quadrilateral with at least one pair of parallel sides

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.

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.

Concyclic points

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.

Isosceles trapezoid Trapezoid symmetrical about an axis

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.

Square Regular quadrilateral

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.

Simson line

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.

Tangential quadrilateral

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.

Varignons theorem The midpoints of the sides of an arbitrary quadrilateral form a 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 was published posthumously in 1731.

Bicentric quadrilateral Type of shape

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.

Ex-tangential 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.

Tangential trapezoid

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.

Equidiagonal 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 to whether they were equidiagonal and then into more specialized types.

Right kite Symmetrical quadrilateral

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.

Newton–Gauss line

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

References

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