Quadrilateral whose vertices can all fall on a single circle
Examples of cyclic quadrilaterals
In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.
The word cyclic is from the Ancient Greek κύκλος (kuklos), which means "circle" or "wheel".
All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus. The section characterizations below states what necessary and sufficient conditions a quadrilateral must satisfy to have a circumcircle.
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In Euclidean geometry, a cyclicquadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called...
diametric quadrilateral is a cyclicquadrilateral having one of its sides as a diameter of the circumcircle. A Hjelmslev quadrilateral is a quadrilateral with...
of quadrilaterals are inscriptable quadrilateral, inscriptible quadrilateral, inscribable quadrilateral, circumcyclic quadrilateral, and co-cyclic quadrilateral...
not necessarily concyclic. After triangles, the special case of cyclicquadrilaterals has been most extensively studied. In general the centre O of a...
certain triangles inside a cyclicquadrilateral are vertices of a rectangle. Triangulating an arbitrary cyclicquadrilateral by its diagonals yields four...
theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclicquadrilateral taken three at a...
five-volume work Treatise on the Quadrilateral. A generalization of the law of tangents holds for a cyclicquadrilateral ◻ A B C D . {\displaystyle \square...
result in geometry is his formula for cyclicquadrilaterals. Given the lengths of the sides of any cyclicquadrilateral, Brahmagupta gave an approximate and...
tangential quadrilateral are given in Tangential quadrilateral#Collinear points. In a cyclicquadrilateral, the circumcenter, the vertex centroid (the intersection...
concyclic points. All triangles are cyclic polygons. Cyclicquadrilateral, a special case of a cyclic polygon. Smallest-circle problem, the related problem...
called the circumscribed circle, is called a cyclic polygon, or in the special case n = 4, a cyclicquadrilateral. All rectangles, isosceles trapezoids, right...
bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclicquadrilaterals. Other names for these quadrilaterals are chord-tangent...
geometry, a harmonic quadrilateral, or harmonic quadrangle, is a quadrilateral that can be inscribed in a circle (cyclicquadrilateral) in which the products...
the sides of the quadrilateral are the vertices of a cyclicquadrilateral. A convex quadrilateral is orthodiagonal if and only if its Varignon parallelogram...
convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side, hence bisecting the latter side. If the quadrilateral is...
bisector. The internal angle bisectors of a convex quadrilateral either form a cyclicquadrilateral (that is, the four intersection points of adjacent...
example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called...
his famous theorem on the diagonals of a cyclicquadrilateral: Brahmagupta's theorem: If a cyclicquadrilateral has diagonals that are perpendicular to...
consequence of the theorem, opposite angles of cyclicquadrilaterals sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in...
concurrencies of a tangential quadrilateral are given here. In a cyclicquadrilateral, four line segments, each perpendicular to one side and passing through...
In geometry, Brahmagupta's theorem states that if a cyclicquadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular...
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