View source for Newton's theorem (quadrilateral)

{{Short description|Geometrical theorem}}
[[File:Newton theorem.svg|thumb|upright=1.5| ''P'' lies on the Newton line ''EF'']]
In [[Euclidean geometry]] '''Newton's theorem''' states that in every [[tangential quadrilateral]] other than a [[rhombus]], the center of the [[incircle]] lies on the [[Newton line]].

== Statement ==

Let ''ABCD'' be a tangential quadrilateral with at most one pair of parallel sides. Furthermore, let ''E'' and ''F'' the midpoints of its diagonals ''AC'' and ''BD'' and ''P'' be the center of its incircle. Given such a configuration the point P is located on the Newton line, that is line ''EF'' connecting the midpoints of the diagonals.{{r|alsina}}

A tangential quadrilateral with two pairs of parallel sides is a rhombus. In this case, both midpoints and the center of the incircle coincide, and by definition, no Newton line exists.

== Proof ==
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''&nbsp;+&nbsp;''c''&nbsp;=&nbsp;''b''&nbsp;+&nbsp;''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 sufficient to ensure that ''P'' lies on ''EF''. Let ''r'' be the radius of the incircle, then ''r'' is also the altitude of all four triangles.{{r|alsina}}
<math display="block">
 \begin{align} 
  A(\triangle PAB) + A(\triangle PCD) &= \frac{1}{2}ra + \frac{1}{2}rc = \frac{1}{2}r(a+c) \\
  &= \frac{1}{2}r(b+d) = \frac{1}{2}rb + \frac{1}{2}rd \\
  &= A(\triangle PBC) + A(\triangle PAD).
\end{align}
</math>

==References==
{{reflist|refs = 

<ref name="alsina">{{cite book
 | last1 = Alsina | first1 = Claudi 
 | last2 = Nelsen | first2 = Roger B.
 | year = 2010
 | title = Charming Proofs: A Journey Into Elegant Mathematics
 | publisher = [[Mathematics Association of America]]
 | isbn = 9780883853481
 | pages = 117–118 
 | url = https://books.google.com/books?id=mIT5-BN_L0oC&pg=PA117
}}</ref>

}}

== External links ==
* [http://www.cut-the-knot.org/Curriculum/Geometry/NewtonTheorem.shtml#explanation ''Newton’s and Léon Anne’s Theorems''] at cut-the-knot.org

[[Category:Theorems about quadrilaterals and circles]]