A similarity system of triangles is a specific configuration involving a set of triangles. A set of triangles is considered a configuration when all of the triangles share a minimum of one incidence relation with one of the other triangles present in the set. An incidence relation between triangles refers to when two triangles share a point. For example, the two triangles to the right, A H C {\displaystyle AHC} and B H C {\displaystyle BHC} , are a configuration made up of two incident relations, since points C {\displaystyle C} and H {\displaystyle H} are shared. The triangles that make up configurations are known as component triangles. Triangles must not only be a part of a configuration set to be in a similarity system, but must also be directly similar. Direct similarity implies that all angles are equal between two given triangle and that they share the same rotational sense. As is seen in the adjacent images, in the directly similar triangles, the rotation of B {\displaystyle B} onto C {\displaystyle C} and B 1 {\displaystyle B^{1}} onto C 1 {\displaystyle C^{1}} occurs in the same direction. In the opposite similar triangles, the rotation of B {\displaystyle B} onto C {\displaystyle C} and B 1 {\displaystyle B^{1}} onto C 1 {\displaystyle C^{1}} occurs in the opposite direction. In sum, a configuration is a similarity system when all triangles in the set, lie in the same plane and the following holds true: if there are n triangles in the set and n − 1 triangles are directly similar, then n triangles are directly similar.
Background J.G. Mauldon introduced the idea of similarity systems of triangles in his paper in Mathematics Magazine "Similar Triangles". Mauldon began his analyses by examining given triangles A B C , X Y Z {\displaystyle ABC,XYZ} for direct similarity through complex numbers, specifically the equation | a b c x y z 1 1 1 | = 0 {\displaystyle {\begin{vmatrix}a&b&c\\x&y&z\\1&1&1\end{vmatrix}}=0} . He then furthered his analyses to equilateral triangles, showing that if a triangle A B C {\displaystyle ABC} satisfied the equation a + w b + w 2 c = 0 {\displaystyle a+wb+w^{2}c=0} when w = − 1 + i √ 3 2 {\displaystyle w={\frac {-1+i\surd 3}{2}}} , it was equilateral. As evidence of this work, he applied his conjectures on direct similarity and equilateral triangles in proving Napoleon's theorem. He then built off Napoleon by proving that if an equilateral triangle was constructed with equilateral triangles incident on each vertex, the midpoints of the connecting lines between the non-incident vertices of the outer three equilateral triangles create an equilateral triangle. Other similar work was done by the French Geometer Thébault in his proof that given a parallelogram and squares that lie on each side of the parallelogram, the centers of the squares create a square. Mauldon then analyzed coplanar sets of triangles, determining if they were similarity systems based on the criterion, if all but one of the triangles were directly similar, then all of the triangles are directly similar.
Examples
Triangles appended to a rectangle
Direct similarity If we construct a rectangle A B C D {\displaystyle ABCD} with directly similar triangles P A B , Q B C , R C D , S D A {\displaystyle PAB,QBC,RCD,SDA} on each side of the rectangle that are similar to P Q S {\displaystyle PQS} , then R Q S {\displaystyle RQS} is directly similar and the set of triangles { P A B , Q B C , R C D , S D A , P Q S , R Q S } {\displaystyle \{PAB,QBC,RCD,SDA,PQS,RQS\}} is a similarity system.
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