Bottema's theorem - Misplaced Pages

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

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

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

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

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

In any given triangle ${\textstyle ABC}$ construct two regular polygons on two sides ${\textstyle AC}$ and ${\textstyle BC}$ . Take the points $D_{1}$ and $D_{2}$ on the circumcircles of the polygons, which are diametrically opposed of the common vertex ${\textstyle C}$ . Then, the midpoint of the line segment $D_{1}D_{2}$ is independent of the location of ${\textstyle C}$ .

References

Koetsier, T. (2007). "Oene Bottema (1901–1992)". In Ceccarelli, M. (ed.). Distinguished Figures in Mechanism and Machine Science. History of Mechanism and Machine Science. Vol. 1. Dordrecht: Springer. pp. 61–68. doi:10.1007/978-1-4020-6366-4_3. ISBN 978-1-4020-6365-7.
Shriki, A. (2011), "Back to Treasure Island", The Mathematics Teacher, 104 (9): 658–664, doi:10.5951/MT.104.9.0658, JSTOR 20876991.
Meskhishvili, M. (2022), "Two Regular Polygons with a Shared Vertex", Communications in Mathematics and Applications, 13 (2): 435–447, arXiv:2206.10374, doi:10.26713/cma.v13i2.1944

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