Bottema's theorem

Short description: Theorem about the midpoint of a line connecting squares on two sides of a triangle

Bottema's theorem construction; changing the location of vertex [math]\displaystyle{ C }[/math] changes the locations of vertices [math]\displaystyle{ E }[/math] and [math]\displaystyle{ F }[/math] but does not change the location of their midpoint [math]\displaystyle{ M }[/math]

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 [math]\displaystyle{ ABC }[/math], construct squares on any two adjacent sides, for example [math]\displaystyle{ AC }[/math] and [math]\displaystyle{ BC }[/math]. The midpoint of the line segment that connects the vertices of the squares opposite the common vertex, [math]\displaystyle{ C }[/math], of the two sides of the triangle is independent of the location of [math]\displaystyle{ C }[/math].

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]\displaystyle{ 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.

See also

• Van Aubel's theorem
• Napoleon's theorem

External links

• Bottema's Theorem: What Is It?
• Wolfram Demonstrations Project – Bottema's Theorem