Median Theorem for Polygons
The median theorem for polygons is a generalization of the median theorem for triangles.[more]
A median of a triangle is a line segment from a vertex to the midpoint of its opposite side.
The median theorem for triangles: The medians of a triangle intersect in a point that is two-thirds of the way from a vertex to the midpoint of its opposite side.
The midpoint and medians of an -gon may be defined inductively.
The midpoint of a 1-gon (a point) is the point itself.
The midpoint of a 2-gon (a line segment) is the midpoint of the line segment, the point equidistant from its endpoints.
A median of an -gon , , is a line segment from the midpoint of a -gon defined by of the vertices of , , and the midpoint of a complementary -gon formed from the other vertices of . A 2-gon has just one median, the 2-gon itself.
The midpoint of an -gon, , is the intersection of any two of its medians.
The median theorem for polygons: The midpoint of an -gon is well defined; its medians all intersect in a point. Its midpoint is of the way from the midpoint of a -subgon to the midpoint of a complementary -subgon.
Proof: Let be an -gon defined by points in a vector space . The midpoint of is , where is the vector sum of the vertices of . This point satisfies the definition.
Let and be complementary and subgons. Let and .
The point of the way from to is
With or , the median theorem is established for polygons in the plane or in three-space.
This Demonstration illustrates the median theorem for polygons with vertices on a circle, for both regular polygons and polygons with randomly chosen sides. Complementary subgons are colored red and blue. If you choose random sides, click "new random sides" to obtain a polygon of your liking. When you change , you need to click at least once to get an -gon.
For a given , you can choose the number of vertices for the red subgon, and the number of complementary subgons and medians to be displayed simultaneously in the right figure. The left figure displays the red/blue subgons with its median, .[less]
R. B. Kirchner, "A Generalization of the Median Theorem for Triangles," The American Mathematical Monthly, 69(7), 1962 p. 650.