In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called...

diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle. A Hjelmslev quadrilateral is a quadrilateral with...

century Indian mathematician, is used to find the area of any convex cyclic quadrilateral (one that can be inscribed in a circle) given the lengths of the...

not necessarily concyclic. After triangles, the special case of cyclic quadrilaterals has been most extensively studied. In general the centre O of a...

of quadrilaterals are inscriptable quadrilateral, inscriptible quadrilateral, inscribable quadrilateral, circumcyclic quadrilateral, and co-cyclic quadrilateral...

that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle. It was originally stated on a sangaku...

of the diagonal triangle of a cyclic quadrilateral. The point of intersection of the bimedians of the cyclic quadrilateral belongs to the nine-point circle...

five-volume work Treatise on the Quadrilateral. A generalization of the law of tangents holds for a cyclic quadrilateral ◻ A B C D . {\displaystyle \square...

relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named...

theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a...

complete quadrilaterals that are associated with cyclic quadrilaterals, based on the work of Barbu and Patrascu. Given any cyclic quadrilateral ABCD, let...

result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and...

special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral). A modern proof, which uses algebra and is quite different from...

concyclic points. All triangles are cyclic polygons. Cyclic quadrilateral, a special case of a cyclic polygon. Smallest-circle problem, the related problem...

tangential quadrilateral are given in Tangential quadrilateral#Collinear points. In a cyclic quadrilateral, the circumcenter, the vertex centroid (the intersection...

convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side, hence bisecting the latter side. If the quadrilateral is...

regarded as a quadrilateral with one side of length zero. From this perspective, as d {\displaystyle d} approaches zero, a cyclic quadrilateral converges...

geometry, a harmonic quadrilateral, or harmonic quadrangle, is a quadrilateral that can be inscribed in a circle (cyclic quadrilateral) in which the products...

called the circumscribed circle, is called a cyclic polygon, or in the special case n = 4, a cyclic quadrilateral. All rectangles, isosceles trapezoids, right...

bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent...

bisector. The internal angle bisectors of a convex quadrilateral either form a cyclic quadrilateral (that is, the four intersection points of adjacent...

K=rs.} The simplest form of Brahmagupta's formula for the area of a cyclic quadrilateral has a form similar to that of Heron's formula for the triangle area:...

triangle Quadrilateral – 4 sides Cyclic quadrilateral Kite Rectangle Rhomboid Rhombus Square (regular quadrilateral) Tangential quadrilateral Trapezoid...

his famous theorem on the diagonals of a cyclic quadrilateral: Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to...

area can be computed using Brahmagupta's formula for the area of a cyclic quadrilateral, which with two sides equal simplifies to K = ( s − c ) ( s − a )...

the sides of the quadrilateral are the vertices of a cyclic quadrilateral. A convex quadrilateral is orthodiagonal if and only if its Varignon parallelogram...

consequence of the theorem, opposite angles of cyclic quadrilaterals sum to 180°; conversely, any quadrilateral for which this is true can be inscribed in...

In geometry, Brahmagupta's theorem states that if a cyclic quadrilateral is orthodiagonal (that is, has perpendicular diagonals), then the perpendicular...

triangulates a cyclic quadrilateral, the sum of inradii of triangles is constant. After proving the quadrilateral case, the general case of the cyclic polygon...

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...