Файл:Bottema's theorem.pngBottema's theorem construction; changing the location of vertex <math display="inline">C</math> changes the locations of vertices <math display="inline">E</math> and <math display="inline">F</math> but does not change the location of their midpoint <math display="inline">M</math>
The theorem can be stated as follows: in any given triangle <math display="inline">ABC</math>, construct squares on any two adjacent sides, for example <math display="inline">AC</math> and <math display="inline">BC</math>. The midpoint of the line segment that connects the vertices of the squares opposite the common vertex, <math display="inline">C</math>, of the two sides of the triangle is independent of the location of <math display="inline">C</math>.[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, <math display="inline">A</math>, 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 <math display="inline">S</math> is the projection of <math display="inline">M</math> onto <math display="inline">AB</math>, Then <math display="inline">AS=BS=MS</math>.
If the squares are replaced by regular polygons of the same type, then a generalized Bottema theorem is obtained: [3]
In any given triangle <math display="inline">ABC</math> construct two regular polygons on two sides <math display="inline">AC</math> and <math display="inline">BC</math>.
Take the points <math>D_1</math> and <math>D_2</math> on the circumcircles of the polygons, which are diametrically opposed of the common vertex <math display="inline">C</math>. Then, the midpoint of the line segment <math>D_1D_2</math> is independent of the location of <math display="inline">C</math>.
