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A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square. [7] A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle. [8] A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices. [9]
Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle. In 1836 Duncan Gregory generalized this result as follows: Given any convex cyclic 2 n -gon, then the two sums of alternate interior angles are each equal to ( n -1) π {\displaystyle \pi } . [ 4 ]
If every internal angle of a simple polygon is less than a straight angle (π radians or 180°), then the polygon is called convex. In contrast, an external angle (also called a turning angle or exterior angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side. [1]: pp. 261–264
A quadrilateral is a square if and only if it is any one of the following: [2] [3]. A rectangle with two adjacent equal sides; A rhombus with a right vertex angle; A rhombus with all angles equal
1.2.3 Inscribed angles with the center of the circle in their exterior. 1.3 ... As a consequence of the theorem, opposite angles of cyclic quadrilaterals sum to ...
It divides the quadrilateral into two congruent triangles that are mirror images of each other. [7] One diagonal bisects both of the angles at its two ends. [7] Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape [10] [11] and which are in turn named for a hovering bird and the sound it makes.
A convex quadrilateral is ex-tangential if and only if there are six concurrent angles 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.
In a tangential quadrilateral, the four angle bisectors meet at the center of the incircle. 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]