I found this problem rather cute, here is a sketch of my solution: The answer is yes. Note that it's enough to prove this for a triangle, since you can triangulate any polygon. Furthermore, every triangle may be cut into isosceles triangles, so it's enough to prove it for an isosceles triangle. Firstly, the right isosceles triangle is done so: (see the first picture). From this, we can also cut an obtuse isosceles triangle (see the second picture). Finally, any acute isosceles triangle can be cut into obtuse isosceles triangles, so we're done. $\square$

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How much partial can one expect on this if shown that dissection of isocoeles triangles. and that construction only for equilateral triangles

In fact, this problem was posted in Kvant (1980 year) as M607...