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 ...

, a right kite is 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 ...

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 polytop ...

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 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) 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 fo ...

of the circumcircle. In a tangential quadrilateral (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.

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 b ...

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 or planar lamina, while '' surface area'' refers to the area of an ope ...

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 ca ...

(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) :

R=\tfrac12\sqrt

and, since all kites are tangential quadrilaterals, 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.

Duality

The dual polygon 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