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In Euclidean geometry Newton's theorem states that in every
''Newton’s and Léon Anne’s Theorems''
at cut-the-knot.org Theorems about quadrilaterals and circles
In Euclidean geometry Newton's theorem states that in every tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the ...
other than a rhombus, the center of the incircle lies on the Newton line
In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides.Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. ...
.
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.
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.
Newton's theorem can easily be derived from Anne's theorem
In Euclidean geometry, Anne's theorem, named after the French mathematician Pierre-Leon Anne (1806–1850) describes an equality of certain areas within a convex quadrilateral.
Specifically, it states:
:''Let be a convex quadrilateral with ...
considering that in tangential quadrilaterals the combined lengths of opposite sides are equal (Pitot theorem
In geometry, the Pitot theorem, named after the French engineer Henri Pitot, states that in a tangential quadrilateral (i.e. one in which a circle can be inscribed) the two sums of lengths of opposite sides are the same. Both sums of lengths equa ...
: ''a'' + ''c'' = ''b'' + ''d''). Now 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.
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References
*Claudi Alsina, Roger B. Nelsen: ''Charming Proofs: A Journey Into Elegant Mathematics''. MAA, 2010, , pp. 117–118 ({{Google books, mIT5-BN_L0oC, online copy, page=117)External links
''Newton’s and Léon Anne’s Theorems''
at cut-the-knot.org Theorems about quadrilaterals and circles