Content deleted Content added
Line 14:
This process can be repeated for ''P''<sub>2</sub> to get ''P''<sub>3</sub> by first adding tiles to each edge of ''P''<sub>2</sub> and then filling in the tiles round each vertex of ''P''<sub>2</sub>. Then the process can be repeated from ''P''<sub>3</sub>, to get ''P''<sub>4</sub> and so on, successively producing ''P''<sub>''n''</sub> from ''P''<sub>''n'' – 1</sub>. It can be checked inductively that these are all convex polygons, with non-overlapping tiles.
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 hence that ''P''<sub>''n''</sub> is a convex polygon.
The equality above for ''a'', ''b'' and ''c''
| |||