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We add a bit to Tiling the plane with pair-wise non-congruent and mutually similar triangles and Cutting polygons into mutually similar and non-congruent pieces

A (non square) rectangle can obviously be cut into 3 mutually similar and pairwise non-congruent right triangles. And it is trivial to construct quadrilaterals formed by two similar but non-congruent and non-right triangles.

  1. Can a square be cut into some finite number of triangles that are (1) mutually similar and (2) non-right? No other constraints.

  2. If the answer to the above is "yes", can a square be cut into some finite number of triangles that are (1) mutually similar, (2) non-right and (3) pairwise non-congruent?

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    $\begingroup$ Wait, why is it obvious that a square can be cut into 3 mutually similar and pairwise non-congruent right triangles? For a non-square rectangle it is obvious, but that method does not work for the square. $\endgroup$
    – Zerox
    Commented Dec 30, 2023 at 16:04
  • $\begingroup$ thanks for the correction. edited! $\endgroup$
    – Nandakumar R
    Commented Dec 30, 2023 at 16:21

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