The middle polygon (the Kazner polygon [1] [2] ) is the polygon whose vertices are the midpoints of the edges of the original polygon [3] [4] .
The middle triangle has the same centroid and the same medians as the original triangle. The perimeter of the middle triangle is equal to the half-perimeter of the original triangle, and the area is equal to a quarter of the area of the original triangle (shown using the Heron formula ). The orthocenter of the middle triangle coincides with the center of the circumscribed circle of the original triangle.
By virtue of the Varignon theorem, the middle quadrilateral is always a parallelogram , which is called the varignon. If the quadrilateral is simple , then the area of the parallelogram is equal to half the area of the original quadrilateral. The perimeter of a parallelogram is equal to the sum of the diagonals of the original quadrilateral.
Notes
- ↑ Kasner, 1903 , p. 59.
- ↑ Schoenberg, 1982 , p. 91, 101.
- ↑ Gardner, 2006 , p. 36.
- ↑ Gardner, Gritzmann, 1999 , p. 92.
Literature
- Richard J. Gardner. Geometric tomography. - 2nd. - Cambridge University Press, 2006. - T. 58. - (Encyclopedia of Mathematics and its Applications).
- Richard J. Gardner, Peter Gritzmann. Discrete tomography: Foundations, Algorithms, and Applications / Gabor T. Herman, Attila Kuba. - Springer, 1999. - S. 85–114.
- Edward Kasner. The Group Generated by Central Symmetries, with Application to Polygons // American Mathematical Monthly . - 1903. - T. 10 , no. 3 (March) . - S. 57–63 . - DOI : 10.2307 / 2968300 .
- IJ Schoenberg. Mathematical time exposures. - Mathematical Association of America , 1982. - ISBN 0-88385-438-4 .
- Elwyn R. Berlekamp, Edgar N. Gilbert, Frank W. Sinden. A Polygon Problem // American Mathematical Monthly . - 1965. - T. 72 , no. 3 (March) . - S. 233–241 . - DOI : 10.2307 / 2313689 .
- JH Cadwell. A Property of Linear Cyclic Transformations // The Mathematical Gazette . - 1953. - T. 37 , no. 320 (May) . - S. 85–89 .
- Richard J. Clarke. Sequences of Polygons // Mathematics Magazine . - 1979. - T. 52 , no. 2 (March) . - S. 102–105 . - DOI : 10.2307 / 2689847 .
- Hallard T. Croft, KJ Falconer, Richard K. Guy. Unsolved Problems in Geometry. - Springer, 1991. - S. 76–78.
- Gaston Darboux. Sur un problème de géométrie élémentaire // Bulletin des sciences mathématiques et astronomiques, Sér. 2. - 1878. - T. 2 , no. 1 . - S. 298-304 .
- Y. David Gau, Lindsay A. Tartre. The Sidesplitting Story of the Midpoint Polygon // Mathematics Teacher. - 1994.- T. 87 , no. 4 (April) . - S. 249–256 .
Links
- Weisstein, Eric W. Midpoint Polygon on Wolfram MathWorld .