Right kite

Special case

A special case of right kites are squares, where the diagonals have equal lengths, and the incircle and circumcircle are concentric.

Characterizations

A kite is a right kite if and only if 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{A}{2}}=\frac{b}{a},\qquad \tan{\frac{C}{2}}=\frac{a}{b}

where a = AB = AD and b = BC = CD. The area of a right kite is :\displaystyle K=ab.

The diagonal AC that is a line of symmetry has the length :p=\sqrt{a^2+b^2}

and, since the diagonals are perpendicular (so a right kite is an orthodiagonal quadrilateral with area K=\frac{pq}{2}), the other diagonal BD has the length :q=\frac{2ab}{\sqrt{a^2+b^2}}.

The radius of the circumcircle is (according to the Pythagorean theorem) :R=\tfrac12\sqrt{a^2+b^2}

and, since all kites are tangential quadrilaterals, the radius of the incircle is given by :r=\frac{K}{s}=\frac{ab}{a+b}

where s is the semiperimeter.

The area is given in terms of the circumradius R and the inradius r as{{citation |access-date=November 1, 2012 |archive-date=December 5, 2022 |archive-url=https://web.archive.org/web/20221205165012/https://forumgeom.fau.edu/FG2012volume12/FG201222.pdf |url-status=dead :K=r(r+\sqrt{4R^2+r^2}).

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. 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

References

1. Michael de Villiers, ''Some Adventures in Euclidean Geometry'', {{ISBN. 978-0-557-10295-2, 2009, pp. 154, 206.
2. [http://www.1728.org/quadkite.htm 1728 Software Systems, ''Kite Calculator'', accessed 8 October 2012]