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If I want to know What's the minimal length of each side of an equilateral triangle that can get a square of which each of its sides is 15 mm, inside it (=inside the square)?. How to calculate it? (I'm not looking for the answer, but I'm looking for the way or approach to solving it, for future self-solutions).

J. W. Tanner
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Arithmetic-Enthusiast
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Draw a pic. You'll want the square as big as possible, so it'll be tangent on a side and touching at two corners. Once you have the pic, think about what angles you know, and what the ratios between sides are in the known shapes (ex: squares have same sides, (30-60-90) triangles have $1-\sqrt 3-2$ sides).

Eric
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  • Each angle is 60 degrees, from here I'm kind of lost... Maybe more hints? – Arithmetic-Enthusiast Apr 06 '21 at 00:42
  • @Arithmetic-Enthusiast There are more than enough hints in this answer to solve the problem. Any more, and Eric would be feeding you the answer. – Rushabh Mehta Apr 06 '21 at 00:42
  • It feels intuitively right that orienting the triangle with one of its sides overlapping a side of the square is what gives the smallest triangle ... but still something that could deserve more of a proof than just pointing at a drawing and saying it's obvious. – Troposphere Apr 06 '21 at 02:03
  • Yeah, I think it’s non-trivial to prove that but it’s intuitively obvious (and the OP just wanted an answer probably for a hs geometry class, so I doubt he cares about a rigorous proof). You can pretty easily make maximality arguments that at least three corners need to touch the sides of the triangle. From there, I would guess it takes some annoying algebra to show that an edge touching is optimal. – Eric Apr 06 '21 at 02:27