Brune's theorem

The theorem of Brune , found and published in 1841 by a Berlin accountant Brune , is a theorem of elementary quadrangular geometry . The theorem deals with and answers the question of how a convex quadrilateral of the Euclidean plane can be constructively divided into four partial quadrilaterals of identical area .

Formulation of the sentence

The sentence can be formulated as follows:

drawn illustration

An arbitrary convex quadrangle of the Euclidean plane is given. On the two diagonals and suppose and are the two center points . ${\displaystyle ABCD}$ ${\displaystyle AC}$ ${\displaystyle BD}$ ${\displaystyle M}$ ${\displaystyle N}$

The point is in the case , so if a parallelogram is the point , while that in the other case, the intersection may be, which results, when going through each of the two diagonal center points and the parallel to each other diagonal pulls . ${\displaystyle O}$ ${\displaystyle M=N}$ ${\displaystyle ABCD}$ ${\displaystyle M}$ ${\displaystyle O}$ ${\displaystyle M}$ ${\displaystyle N}$

Then:

Connects to the point with the centers of the four sides of the quadrangle, the quadrangle is divided into four sub-rectangles, the area of each of the surface area of accounts. ${\displaystyle O}$ ${\displaystyle {\tfrac {1}{4}}}$ ${\displaystyle ABCD}$

literature

• Brune: A property of the quadrangle . In: Crelles Journal . tape 22 , 1841, pp. 379 ( [1] - MR1578286 ).
• Friedrich Joseph Pythagoras Riecke (Hrsg.): Mathematische Unterhaltungen . First issue. Dr. Martin Sendet, Walluf near Wiesbaden 1973, ISBN 3-500-26010-1 (unchanged reprint of the Stuttgart edition 1867–1873).

Individual evidence and notes

1. ↑ Possibly EW Brune after Maximilian Simon , About the development of elementary geometry in the 19th century, Jb DMV, 1st supplementary volume, 1906, p. 256 (register). EW Brune is also known as a pioneer of life tables in Germany (Crelle J. 1837, p. 58).
2. ↑ ^{a b} Friedrich Joseph Pythagoras Riecke (Ed.): Mathematische Unterhaltungen. First issue. 1973, p. 66
3. ↑ See article about Riecke on Wikisource