Bottema's theorem

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 that is linked to the Dutch mathematician Oene Bottema (Groningen, 1901–1992).^[1] As Bottema points out himself, the theorem was known before he published the simple proof.^[2] The theorem is, for example, stated in the popular science book "One, Two, Three... Infinity" by theoretical physicist George Gamow.

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 ${\textstyle C}$.^[3][4]

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: ^[5]

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

See also

• Van Aubel's theorem
• Napoleon's theorem

References

1. Koetsier, Teun (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.
2. Bottema, Oene (1959–1960). "Verscheidenheden XLII. Nogmaals het probleem van de verloren schat" (PDF). Nieuw Tijdschrift voor Wiskunde (in Dutch). 35 (3): 97.
3. Bottema, Oene (1958–1959). "Verscheidenheden XXXVIII. Het probleem van de verloren schat" (PDF). Nieuw Tijdschrift voor Wiskunde (in Dutch). 34 (7): 210.
4. Shriki, Atara (2011), "Back to Treasure Island", The Mathematics Teacher, 104 (9): 658–664, doi:10.5951/MT.104.9.0658, JSTOR 20876991.
5. Meskhishvili, Mamuka (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

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