Search results
Results From The WOW.Com Content Network
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
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.
One of the triangle area formulas involving the semiperimeter also applies to tangential quadrilaterals, which have an incircle and in which (according to Pitot's theorem) pairs of opposite sides have lengths summing to the semiperimeter—namely, the area is the product of the inradius and the semiperimeter: =.
By the secant-tangent theorem, the square of this tangent length equals the power of the point P in the circle C. This power equals the product of distances from P to any two intersection points of the circle with a secant line passing through P. The angle θ between a chord and a tangent is half the arc belonging to the chord.
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 .
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, 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 ...