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In 
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the midpoint polygon of a polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
is the polygon whose vertices are the midpoints of the edges of . It is sometimes called the Kasner polygon after Edward Kasner
Edward Kasner (April 2, 1878 – January 7, 1955) was an American mathematician who was appointed Tutor on Mathematics in the Columbia University Mathematics Department. Kasner was the first Jewish
Jews ( he, יְהוּדִים, , ...
, who termed it the ''inscribed polygon'' "for brevity".
Examples
Triangle
The midpoint polygon of atriangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
is called the medial triangle. It shares the same centroid and medians
The Medes ( Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, th ...
with the original triangle. The perimeter of the medial triangle equals the semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
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 itsVarignon parallelogram
Varignon's theorem is a statement in Euclidean geometry, that deals with the construction of a particular parallelogram, the Varignon parallelogram, from an arbitrary quadrilateral (quadrangle). It is named after Pierre Varignon, whose proof wa ...
. If the quadrilateral is simple
Simple or SIMPLE may refer to:
*Simplicity, the state or quality of being simple
Arts and entertainment
* ''Simple'' (album), by Andy Yorke, 2008, and its title track
* "Simple" (Florida Georgia Line song), 2018
* "Simple", a song by Johnn ...
, 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
In geometry, the midpoint-stretching polygon of a cyclic polygon is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the circular arcs between the vertices of .. It may be derived from the midp ...
* Varignon's theorem
References
* * * *Further reading
* * * * * *External links
*{{MathWorld , urlname=MidpointPolygon , title=Midpoint Polygon Polygons