Original question: Construct any regular (or similar-scaled to a given) geometric shape that has the same area as given triangle?

My idea is application of generalized Pythagora's theorem. Euclid Elements Book VI. $31$,

I know it's possible to construct such square and equilateral triangle, but are there some other creative ideas?

I'm looking for a marked ruler and compass construction, or a construction using two (marked) triangles and compass because it really doesn't matter in sense of problem solvability (is that the word?)

EDIT: Thinking thinking, Euclid Elements Book VI. $18$, $\Rightarrow$ Every (regular) polygon can be visualized as multiple triangles

And, after that apply Euclid Elements Book VI. $25$, although I'm not sure if there exist a simpler way to construct a triangle that has the same area and is similar to another given triangle ? (every center triangle of a polygon is equal and has $\frac{1}{n}$ Area of an original triangle)

the answer is obviously long, so I'm not looking for complete answer, just for $1$ tranfsormation of triangle into regular $n$-gon. $n>4$

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    $\begingroup$ If you mean with straightedge and compass, the half circle is impossible because $\pi$ is not constructible. $\endgroup$ – Gae. S. May 27 at 21:34
  • $\begingroup$ Thanks, I'll edit the question. Silly me! $\endgroup$ – bcurious May 27 at 21:36

You can construct any regular polygon with 3 or more sides if you know the area, A, of the triangle you are trying to construct a similar scaled geometric shape from. Using $A$, create a regular polygon with n sides of side length:

$s = \sqrt{\frac{4A\tan(\frac{180}{n})}{n}}$

and with interior angle being:

$a = 180 - \frac{360}{n}$

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  • $\begingroup$ interesting formula, but i think it is not constructible with straightedge and compass. I edited the question according to your answer. $\endgroup$ – bcurious May 28 at 5:34
  • $\begingroup$ Using a pencil, straight edge, and a compass you technically can because you know side lengths and angles. But if you want to use the original triangle as a basis then yeah this is different $\endgroup$ – justaguy May 28 at 7:27
  • $\begingroup$ area A is not given implicitly $\endgroup$ – bcurious May 28 at 8:19
  • $\begingroup$ I don't agree with this. For instance, the angle $\frac{360^\circ}7$ is provably impossible to contruct with compass and straightedge. $\endgroup$ – Gae. S. May 28 at 14:54
  • $\begingroup$ This is fair, if you really want me to I can limit this answer to polygons with nonrepeating rational values for angles and sidelengths $\endgroup$ – justaguy May 28 at 15:47

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