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A self-intersecting quadrilateral is called variously a cross-quadrilateral, crossed quadrilateral, butterfly quadrilateral or bow-tie quadrilateral. In a crossed quadrilateral, the four "interior" angles on either side of the crossing (two acute and two reflex, all on the left or all on the right as the figure is traced out) add up to 720°. [10]
Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid: It has two adjacent angles that are supplementary, that is, they add up to 180 degrees. The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.
The cyclic quadrilaterals may equivalently defined as the quadrilaterals in which two opposite angles are supplementary (they add to 180°); if one pair is supplementary the other is as well. [9] Therefore, the right kites are the kites with two opposite supplementary angles, for either of the two opposite pairs of angles.
Lambert quadrilateral fundamental domain in orbifold *p222 *3222 symmetry with 60-degree angle on one of its corners. *4222 symmetry with 45-degree angle on one of its corners. The limiting Lambert quadrilateral has three right angles, and one 0-degree angle with an ideal vertex at infinity, defining orbifold *∞222 symmetry.
Conversely, a convex quadrilateral in which the four angle bisectors meet at a point must be tangential and the common point is the incenter. [4] According to the Pitot theorem, the two pairs of opposite sides in a tangential quadrilateral add up to the same total length, which equals the semiperimeter s of the quadrilateral:
To prove that AF = FM, first note that the angles FAM and CBM are equal, because they are inscribed angles that intercept the same arc of the circle (CD). Furthermore, the angles CBM and CME are both complementary to angle BCM (i.e., they add up to 90°), and are therefore equal. Finally, the angles CME and FMA are the same.
In Euclidean geometry, a right kite is a kite (a quadrilateral 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. [1] That is, it is a kite with a circumcircle (i.e., a cyclic kite). Thus the right kite is a convex quadrilateral and has two opposite right ...
In Euclidean geometry, the measures of the interior angles of a triangle add up to π radians, 180°, or 1 / 2 turn; the measures of the interior angles of a simple convex quadrilateral add up to 2 π radians, 360°, or 1 turn.
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