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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]
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
Download as PDF; Printable version; ... of a convex quadrilateral intersect in a point that lies on the Newton ... If the quadrilateral is a tangential quadrilateral, ...
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 ...
Download as PDF; Printable version; In other projects Wikidata item; ... square (regular quadrilateral) Tangential quadrilateral; Trapezoid or trapezium Isosceles ...
Additionally, if a convex kite is not a rhombus, there is a circle outside the kite that is tangent to the extensions of the four sides; therefore, every convex kite that is not a rhombus is an ex-tangential quadrilateral. The convex kites that are not rhombi are exactly the quadrilaterals that are both tangential and ex-tangential. [17]
The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices. All triangles are tangential, as are all regular polygons with any number of sides. A well-studied group of tangential polygons are the tangential quadrilaterals, which include the rhombi and kites.