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

, a right kite is a

kite A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...

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, ...

whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other) that can be inscribed in a circle.Michael de Villiers, ''Some Adventures in Euclidean Geometry'', , 2009, pp. 154, 206. That is, it is a kite with a

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 ...

(i.e., 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 ...

kite). Thus the right kite is 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 polytope ...

quadrilateral and has two opposite

right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...

s. If there are exactly two right angles, each must be between sides of different lengths. All right kites are

bicentric quadrilateral 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'' r ...

s (quadrilaterals with both a circumcircle and an incircle), since all kites have an

incircle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

. One of the diagonals (the one that is a line of

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...

) divides the right kite into two

right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...

s and is also a

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 for ...

of the circumcircle. In a

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 ...

(one with an incircle), the four line segments between the center of the incircle and the points where it is tangent to the quadrilateral partition the quadrilateral into four right kites.

Special case

A special case of right kites are

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 ...

s, where the diagonals have equal lengths, and the incircle and circumcircle are

concentric In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...

Characterizations

A kite is a right kite

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 bicondi ...

it has a circumcircle (by definition). This is equivalent to its being a kite with two opposite right angles.

Metric formulas

Since a right kite can be divided into two right triangles, the following metric formulas easily follow from well known properties of right triangles. In a right kite ''ABCD'' where the opposite angles ''B'' and ''D'' are right angles, the other two angles can be calculated from :

\tan=\frac,\qquad \tan=\frac

where ''a'' = ''AB'' = ''AD'' and ''b'' = ''BC'' = ''CD''. The

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 A shape or figure is a graphics, graphical representation of an obje ...

of a right kite is :

\displaystyle K=ab.

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� ...

''AC'' that is a line of symmetry has the length :

p=\sqrt

and, since the diagonals are

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

(so a right kite is an

orthodiagonal 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 ...

with area

K=\frac

), the other diagonal ''BD'' has the length :

q=\frac.

The

radius 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 is (according to 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 opposite t ...

) :

R=\tfrac12\sqrt

and, since all kites are

s, the radius of the incircle is given by :

r=\frac=\frac

where ''s'' is the semiperimeter. The area is given in terms of the circumradius ''R'' and the inradius ''r'' as. :

K=r(r+\sqrt).

If we take the segments extending from the intersection of the diagonals to the vertices in clockwise order to be

d_1

d_2

d_3

, and

d_4

, then, :

d_1 d_3=d_2 d_4

This is a direct result of the

geometric mean theorem The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states ...

Duality

The

dual polygon In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other. Properties Regular polygons are self-dual. The dual of an isogonal (vertex-transitive) polygon is an isotoxal (ed ...

to a right kite is an isosceles tangential trapezoid.

Alternative definition

Sometimes a right kite is defined as a kite with at least one right angle.1728 Software Systems, ''Kite Calculator'', accessed 8 October 2012
/ref> If there is only one right angle, it must be between two sides of equal length; in this case, the formulas given above do not apply.

References

{{Polygons Types of quadrilaterals