Newton's theorem

P lies on the Newton line EF

The set of Newton , named after Isaac Newton (1643-1727) is a statement in the elementary geometry , which means that the center of the inscribed circle of a tangent rectangle on the straight line connecting its two center points of the diagonal is located.

Let ABCD be a tangent quadrilateral with at most one parallel pair of sides and diagonals AC and BD . Furthermore, let E and F be the center points of the diagonals, then the center of the inscribed circle P lies on the straight line connecting E and F , the so-called Newton line .

Newton's theorem can be proved quite easily using Anne and Pitot's theorems . According to Pitot's theorem, the sums of the lengths of the sides of opposite sides of a tangent quadrilateral are the same, i.e. a  +  c  =  b  +  d . If one chooses any point P inside the tangent quadrilateral and then divides it into the triangles PAB , PBC , PCD and PDA , then Anne's theorem, together with its inversion, says that the sum of the areas of the opposite triangles is exactly the same, if they are on the Newton line . Let P be the center of the inscribed circle and r its radius, then:

${\displaystyle {\begin{aligned}&A(\triangle PAB)+A(\triangle PCD)\\=&{\tfrac {1}{2}}ra+{\tfrac {1}{2}}rc={\tfrac {1}{2}}r(a+c)\\=&{\tfrac {1}{2}}r(b+d)={\tfrac {1}{2}}rc+{\tfrac {1}{2}}rd\\=&A(\triangle PBC)+A(\triangle PAD)\end{aligned}}}$

Thus, according to Anne's theorem, P lies on the Newton line.

literature

• Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics . MAA, 2010, ISBN 9780883853481 , pp. 117–118 ( excerpt (Google) )

Web links

• Alexander Bogomolny: Newton's and Léon Anne's Theorems on cut-the-knot.org