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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 ...
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In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
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 .
The line segments GH and IJ that connect the midpoints of opposite sides (the bimedians) of a convex quadrilateral intersect in a point that lies on the Newton line.This point K bisects the line segment EF that connects the diagonal midpoints.
In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. [1] It has also been called an exscriptible quadrilateral. [2] The circle is called its excircle, its radius the exradius and its center the excenter (E in the figure). The ...