Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...

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 polygon, convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This cir ...

other than a

rhombus In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...

, the center of the

incircle 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 ...

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. Properties The line segments and that connect the midpoints of opposite sides (the ...

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. 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 In Euclidean geometry, Anne's theorem describes an equality of certain areas within a convex quadrilateral. This theorem is named after the French mathematician Pierre-Léon Anne (1806–1850). Statement The theorem is stated as follows: Let ...

considering that in tangential quadrilaterals the combined lengths of opposite sides are equal (

Pitot theorem The Pitot theorem in geometry states that in a tangential quadrilateral the two pairs of opposite sides have the same total length. It is named after French engineer Henri Pitot. Statement and converse A tangential quadrilateral is usually defin ...

: ''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 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.

\begin 
  A(\triangle PAB) + A(\triangle PCD) &= \fracra + \fracrc = \fracr(a+c) \\
  &= \fracr(b+d) = \fracrb + \fracrd \\
  &= A(\triangle PBC) + A(\triangle PAD).
\end

References

{{reflist, refs = {{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

External links

''Newton’s and Léon Anne’s Theorems''
at cut-the-knot.org Theorems about quadrilaterals and circles