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Figure 9
We are given a triangle T of minimum size larger than 2k. From one of its vertices (we have taken it as the top one here) we fit in the largest sub-triangle possible (at the very least, there is always a triangle of level 0 that can be fitted in). Its lower side is indicated in green. It must turn inwards at the sides of T and complete to the opposite green triangle. By the induction assumption its other two sides also complete to opposite pairs, and this leads to the new black triangle with the green triangle nested in it. Since we started from a maximal-sized sub-triangle, this larger black triangle must in fact be the entirety of our original triangle T. This shows that T has edge length 2k+1. The visible nesting and the induction hypotheses show that the new triangle is nested within.

ISSN: 2053-2733
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