Midpoint polygon
In geometry, the midpoint polygon of a polygon P is the polygon whose vertices are the midpoints of the edges of P.[1][2] It is sometimes called the Kasner polygon after Edward Kasner, who termed it the inscribed polygon "for brevity".[3][4]
Examples
Triangle
The midpoint polygon of a triangle is called the medial triangle. It shares the same centroid and medians with the original triangle. The perimeter of the medial triangle equals the semiperimeter of the original triangle, and the area is one quarter of the area of the original triangle. This can be proven by the midpoint theorem of triangles and Heron's formula. The orthocenter of the medial triangle coincides with the circumcenter of the original triangle.
Quadrilateral
The midpoint polygon of a quadrilateral is a parallelogram called its Varignon parallelogram. If the quadrilateral is simple, the area of the parallelogram is one half the area of the original quadrilateral. The perimeter of the parallelogram equals the sum of the diagonals of the original quadrilateral.
References
- Gardner 2006, p. 36.
- Gardner & Gritzmann 1999, p. 92.
- Kasner 1903, p. 59.
- Schoenberg 1982, pp. 91, 101.
- Gardner, Richard J. (2006), Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58 (2nd ed.), Cambridge University Press
- Gardner, Richard J.; Gritzmann, Peter (1999), "Uniqueness and Complexity in Discrete Tomography", in Herman, Gabor T.; Kuba, Attila (eds.), Discrete tomography: Foundations, Algorithms, and Applications, Springer, pp. 85–114
- Kasner, Edward (March 1903), "The Group Generated by Central Symmetries, with Application to Polygons", American Mathematical Monthly, 10 (3): 57–63, doi:10.2307/2968300, JSTOR 2968300
- Schoenberg, I. J. (1982), Mathematical time exposures, Mathematical Association of America, ISBN 0-88385-438-4
Further reading
- Berlekamp, Elwyn R.; Gilbert, Edgar N.; Sinden, Frank W. (March 1965), "A Polygon Problem", American Mathematical Monthly, 72 (3): 233–241, doi:10.2307/2313689, JSTOR 2313689
- Cadwell, J. H. (May 1953), "A Property of Linear Cyclic Transformations", The Mathematical Gazette, 37 (320): 85–89, doi:10.2307/3608930, JSTOR 3608930
- Clarke, Richard J. (March 1979), "Sequences of Polygons", Mathematics Magazine, 52 (2): 102–105, doi:10.2307/2689847, JSTOR 2689847
- Croft, Hallard T.; Falconer, K. J.; Guy, Richard K. (1991), "B25. Sequences of polygons and polyhedra", Unsolved Problems in Geometry, Springer, pp. 76–78
- Darboux, Gaston (1878), "Sur un problème de géométrie élémentaire", Bulletin des sciences mathématiques et astronomiques, Série 2, 2 (1): 298–304
- Gau, Y. David; Tartre, Lindsay A. (April 1994), "The Sidesplitting Story of the Midpoint Polygon", Mathematics Teacher, 87 (4): 249–256, doi:10.5951/MT.87.4.0249