In geometry, the midpoint polygon of a polygonP 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.
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
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