@MISC{Beeson10tilingtriangle, author = {Michael Beeson}, title = {Tiling triangle ABC with congruent triangles similar to ABC}, year = {2010} }

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Abstract

We investigate the problem of cutting a triangle ABC into N congruent triangles (the “tiles”), each of which is similar to ABC. The more general problem when the tile is not similar to ABC is not treated in this paper; see [1]. We give a complete characterization of the numbers N for which some triangle ABC can be tiled by N tiles similar to ABC, and also a complete characterization of the numbers N for which a particular triangle ABC can so tiled. It has long been known that there is always a “quadratic tiling ” when N is a square. We show that unless ABC is a right triangle, N must be a square. On the other hand, if ABC is a right triangle, there are two more possibilities: N can be a sum of two squares e 2 + f 2 if the tangent of one of the angles is the rational number e/f, or in case ABC is a 30-60-90 triangle, N can be three times a square. The key idea is that the similarity factor √ N is an eigenvalue of a certain matrix. The proofs we give involve only undergraduate level linear algebra. 1 Examples of Tilings We consider the problem of cutting a triangle into N congruent triangles. Figures 1 through 4 show that, at least for certain triangles, this can be done with N = 3, 4, 5, 6, 9, and 16. Such a configuration is called an N-tiling. Figure 1: Two 3-tilings The method illustrated for N = 4,9, and 16 clearly generalizes to any perfect square N. While the exhibited 3-tiling, 6-tiling, and 5-tiling clearly depend on the exactly angles of the triangle, any triangle can be decomposed into n 2 congruent triangles by drawing n − 1 lines, parallel to each edge and dividing the other two edges into n equal parts. Moreover, the large (tiled) triangle is similar to the small triangle (the “tile”). We call such a tiling a quadradtic tiling. It follows that if we have a tiling of a triangle ABC into N congruent triangles, and m is any integer, we can tile ABC into Nm 2 triangles by subdividing the first tiling, replacing