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First question below had appeared in a note at Triangles that can be cut into mutually congruent and non-convex polygons

Following the results of Beeson quoted in the answer at Subdivision of triangles into congruent triangles,

  1. Is there any triangle that can be divided into 7 (or 11 or...) connected but not necessarily convex polygons that are mutually congruent?

  2. Will Beeson's impossibility results remain valid if we need the triangular pieces only to be mutually similar (not necessarily similar to the big triangle) rather than congruent?

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    $\begingroup$ Any right triangle T can be divided into two triangles similar to T (by the construction that yields the Pythagorean theorem by comparing areas). Inductively it follows that for each n there is a division of T into n triangles similar to T. $\endgroup$
    – Noam D. Elkies
    Commented May 11 at 12:16
  • $\begingroup$ Thank you for clarifying this - indeed, a right triangle T can be cut into n right triangles similar to itself - for any n. If one needs the pieces to be mutually similar but not similar to T, it might not be possible for all n. $\endgroup$
    – Nandakumar R
    Commented May 11 at 13:00
  • $\begingroup$ To have the pieces similar to each other but not similar to T, start from an isosceles triangle T' obtained by gluing two copies of T along a side. That gives n=2, and for larger n decompose one (or both) copies of T as before to decompose T' into n triangles each similar to T. $\endgroup$
    – Noam D. Elkies
    Commented May 11 at 14:40
  • $\begingroup$ Thanks again. That simply buries question 2 as basically trivial. only q-1 remains. $\endgroup$
    – Nandakumar R
    Commented May 11 at 16:01

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