Bottema's theorem



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 $ABC$, construct squares on any two adjacent sides, for example $AC$ and $BC$. The midpoint of the line segment that connects the vertices of the squares opposite the common vertex, $C$ , of the two sides of the triangle is independent of the location of $C$.

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, $A$, 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 $S$ is the projection of $M$  onto $AB$, Then $AS=BS=MS$.

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

In any given triangle $ABC$ construct two regular polygons on two sides $AC$  and  $BC$. Take the points $$D_1$$  and  $$D_2$$  on the circumcircles of the polygons, which are diametrically opposed of the common vertex $C$. Then, the midpoint of the line segment $$D_1D_2$$  is independent of the location of $C$.