Bottema's theorem

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

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 $ABC$, 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 $D_1{D}_2$ is independent of the location of $C$.

See also

• 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

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