Bottema's theorem

Bottema's theorem construction; changing the location of vertex ${\textstyle C}$ changes the locations of vertices ${\textstyle E}$ and ${\textstyle F}$ but does not change the location of their midpoint ${\textstyle M}$

Bottema's theorem is a theorem in plane geometry by the Dutch mathematician Oene Bottema (Groningen, 1901–1992).

The theorem can be stated as follows: in any given triangle ${\textstyle ABC}$, construct squares on any two adjacent sides, for example ${\textstyle AC}$ and ${\textstyle BC}$. The midpoint of the line segment that connects the vertices of the squares opposite the common vertex, ${\textstyle C}$, of the two sides of the triangle is independent of the location of ${\textstyle C}$.

The theorem is true when the squares are constructed in one of the following ways:

• Looking at the figure, starting from the lower left vertex, ${\textstyle A}$, follow the triangle vertices clockwise and construct the squares to the left of the sides of the triangle.
• Follow the triangle in the same way and construct the squares to the right of the sides of the triangle.

If the squares are replaced by regular polygons of the same type, then a generalized Bottema theorem is obtained:

In any given triangle ${\textstyle ABC}$ construct two regular polygons on two sides ${\textstyle AC}$ and ${\textstyle BC}$. Take the points $D_{1}$ and $D_{2}$ on the circumcircles of the polygons, which are diametrically opposed of the common vertex ${\textstyle C}$. Then, the midpoint of the line segment $D_{1}D_{2}$ is independent of the location of ${\textstyle C}$.

• Van Aubel's theorem
• Napoleon's theorem

External links

• Bottema's Theorem: What Is It?
• Wolfram Demonstrations Project – Bottema's Theorem
• GeoGebra Demonstrations Project - A Generalized Theorem - Equilateral Triangles
• GeoGebra Demonstrations Project - A Generalized Theorem - Regular Pentagons