Bottema's theorem

rdf:langString Bottema's theorem

rdf:langString Bottema's theorem is a theorem in plane geometry by the Dutch mathematician (Groningen, 1901–1992). The theorem can be stated as follows: in any given triangle , construct squares on any two adjacent sides, for example and . The midpoint of the line segment that connects the vertices of the squares opposite the common vertex, , of the two sides of the triangle is independent of the location of . 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, , 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 the squares are replaced by regular polygons of the same type, then a generalized Bottema theorem is obtained: In any given triangle construct two regular polygons on two sides and .Take the points and on the circumcircles of the polygons, which are diametrically opposed of the common vertex . Then, the midpoint of the line segment is independent of the location of . * Van Aubel's theorem * Napoleon's theorem