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The original triangle Δ gives a convex polygon ''P''<sub>1</sub> with 3 vertices. At each of the three vertices the triangle can be successively reflected through edges emanating from the vertices to produce 2''m'' copies of the triangle where the angle at the vertex is {{pi}}/''m''. The triangles do not overlap except at the edges, half of them have their orientation reversed and they fit together to tile a neighborhood of the point. The union of these new triangles together with the original triangle form a connected shape ''P''<sub>2</sub>. It is made up of triangles which only intersect in edges or vertices, forms a convex polygon with all angles less than {{pi}} and each side being the edge of a reflected triangle. This can be seen more clearly by noting that some triangles or tiles are added twice, the three which have a side in common with the original triangle. The rest have only a vertex in common. A more systematic way of performing the tiling is first to add a tile to each side (the reflection of the triangle in that edge) and then fill in the gaps at each vertex. This results in a total of 3 + (2''a'' – 3) + (2''b'' - 3) + (2''c'' - 3) = 2(''a'' + ''b'' + ''c'') - 6 new triangles. The new vertices are of two two types. Those which are vertices of the triangles attached to sides of the original triangle, which are connected to 2 vertices of Δ. Each of these lie in three new triangles which intersect at that vertex. The remainder are connected to a unique vertex of Δ and belong to two new triangles which have a common edge. Thus there are 3 + (2''a'' – 4) + (2''b'' - 4) + (2''c'' - 4) = 2(''a'' + ''b'' + ''c'') - 9 new vertices. By construction there is no overlapping. To see that ''P''<sub>2</sub> is convex, it suffices to see that the angle between sides meeting at a new vertex make an angle less than {{pi}}. But the new vertices lies in two or three new triangles, which meet at that vertex, so the angle at that vertex is less than 2{{pi}}/3 or {{pi}}, as required.
This process can be repeated for ''P''<sub>2</sub> to get ''P''<
Indeed as in the first step of the process there are two types of tile in building ''P''<sub>''n''</sub> from ''P''<sub>''n'' – 1</sub>, those attached to an edge of ''P''<sub>''n'' – 1</sub> and those attached to a single vertex. Similarly there are two types of vertex, one in which two new tiles meet and those in which three tiles meet. So provided that no tiles overlap the previous argument shows that angles at vertices do not exceed {{pi}} and implies that ''P''<sub>''n''</sub> is a convex polygon.
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