A square is drawn on a piece of paper. How can you draw an equilateral triangle such that its vertices lie on the boundary of this square?

  • $\begingroup$ Not too difficult but rather pretty! $\endgroup$ Commented Nov 8, 2020 at 8:50

1 Answer 1


Choose a point on the boundary and rotate the square around this point by 60° cw and ccw. The intersection points with the original square are the vertices of the triangle: ![![enter image description here


The new intersection points (those that are not the pivot point) are 60° rotations of each other wrt to the pivot point because they are the same intersection of two shapes that are simultaneously rotated, namely, blue and black square into black and red square.

  • $\begingroup$ Very nice. This is better than my solution. By the way, you can use this to find the exact location of each vertex of the triangle. $\endgroup$ Commented Nov 8, 2020 at 8:50
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    $\begingroup$ It's not clear what physical actions you're supposed to take with the piece of paper to do this. We can't rotate the paper 60 degrees and look at the intersection, because there won't be two intersecting squares. There'll just be one rotated square. Plus, it's not clear how we would make sure the rotation is 60 degrees. Even if we assume we have a protractor or something, it's going to slip and shift if we rotate the paper under it. $\endgroup$ Commented Nov 8, 2020 at 17:16
  • $\begingroup$ @user2357112supportsMonica Finding and rotating by a 60° angle as well as intersecting basic shapes are standard operations in the majority of contexts that come to mind (straight edge and compass, origami, analytic geometry). Given that it seems eminently justifiable to gloss over these implementation details in both the question and the answer. One may even argue that this generality adds to their appeal. $\endgroup$ Commented Nov 8, 2020 at 22:15
  • $\begingroup$ @user2357112supportsMonica the best tool for this is a second sheet of paper. $\endgroup$ Commented Nov 9, 2020 at 4:16

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