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]
The medial triangle
The Varignon parallelogram

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.
See also

Circulant matrix
Midpoint-stretching polygon
Varignon's theorem

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 58 (2nd ed.), Cambridge University Press
Gardner, Richard J.; Gritzmann, Peter (1999), "Uniqueness and Complexity in Discrete Tomography", in Herman, Gabor T.; Kuba, Attila, 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
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
Cadwell, J. H. (May 1953), "A Property of Linear Cyclic Transformations", The Mathematical Gazette 37 (320): 85–89
Clarke, Richard J. (March 1979), "Sequences of Polygons", Mathematics Magazine 52 (2): 102–105, doi:10.2307/2689847
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, Sér. 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

External links

Weisstein, Eric W., "Midpoint Polygon", MathWorld.

Mathematics Encyclopedia