Cyclic quadrilateral

A cyclic quadrilateral is a quadrilateral whose vertices lie on a circle, the radius of the quadrangle. Consequently, all sides of the quadrilateral tendon tendons of the perimeter. Usually, one thinks of a cyclic quadrilateral does not have proposed cyclic quadrilateral, this is necessarily convex.

Sentences about tendons squares (selection)

The cyclic quadrilateral ABCD is denoted by.

The following rates apply only for non- ailing tendons quadrilaterals ABCD:

The former area formula is a generalization of Heron'schen area formula for triangles and is also referred to as a set of Brahmagupta or formula of Brahmagupta. Here we group a triangle as a degenerate cyclic quadrilateral, whose fourth side has the length 0, ie two of its vertices lie on one another.

A quadrilateral ( and any other polygon also ) has fixed, orderly side lengths if and only the largest possible area, if it is a cyclic quadrilateral (or polygon ).

Opposite angles in cyclic quadrilateral

In the cyclic quadrilateral, the angle sum of the opposite angles is 180 °.

The proof follows directly from the circle angle theorem, since two opposite angles of the quadrilateral are tendons circumferential angle on two complementary arcs whose central angle add up to 360 °. Since circumferential angle is half as large as the central angle over the same arc, the circumferential angles must add up to 360 ° / 2 = 180 °.

Another proof can be found in the proof archive.

Related squares

A cyclic quadrilateral, which is trapezoidal simultaneously, ie isosceles trapezoid. Each rectangle is an isosceles trapezoid and thus a cyclic quadrilateral.

A quadrilateral has an inscribed circle, ie tangent quadrilateral.