Bottema's theorem

Bottema's theorem is a theorem in plane geometry by the Dutch mathematician Oene Bottema (Groningen, 1901–1992).[1]

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 ${\textstyle M}$

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 ${\textstyle C}$.[2]

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.