Bottema's theorem explained

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

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 $C$.^[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, $A$, 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 $C$.

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

Notes and References

1. Book: Koetsier. Distinguished Figures in Mechanism and Machine Science.. Springer. 2007. 978-1-4020-6365-7. Ceccarelli. M.. History of Mechanism and Machine Science. 1. Dordrecht. 61–68. Oene Bottema (1901–1992). 10.1007/978-1-4020-6366-4_3.
2. .