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The incenter of a tangential quadrilateral lies on its Newton line (which connects the midpoints of the diagonals). [22]: Thm. 3 The ratio of two opposite sides in a tangential quadrilateral can be expressed in terms of the distances between the incenter I and the vertices according to [10]: p.15
A tangential quadrilateral is usually defined as a convex quadrilateral for which all four sides are tangent to the same inscribed circle. Pitot's theorem states that, for these quadrilaterals, the two sums of lengths of opposite sides are the same. Both sums of lengths equal the semiperimeter of the quadrilateral. [2]
Newton's theorem can easily be derived from Anne's theorem considering that in tangential quadrilaterals the combined lengths of opposite sides are equal (Pitot theorem: a + c = b + d). According to Anne's theorem, showing that the combined areas of opposite triangles PAD and PBC and the combined areas of triangles PAB and PCD are equal is ...
In Euclidean geometry, Brahmagupta's formula, named after the 7th 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 sides. Its generalized version, Bretschneider's formula, can be used with non-cyclic quadrilateral.
Download as PDF; Printable version; In other projects ... If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line. [3] See also
An equilateral triangle A bicentric kite A bicentric isosceles trapezoid A regular pentagon. In geometry, a bicentric polygon is a tangential polygon (a polygon all of whose sides are tangent to an inner incircle) which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon.
The ex-tangential quadrilateral is closely related to the tangential quadrilateral (where the four sides are tangent to a circle). Another name for an excircle is an escribed circle, [3] but that name has also been used for a circle tangent to one side of a convex quadrilateral and the extensions of the adjacent two sides. In that context all ...
Other names for these quadrilaterals are chord-tangent quadrilateral [1] and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral [ 2 ] and double scribed quadrilateral .