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Regular polygon = all angles have the same measure AND all sides have equal length.

So, is there a possibility to draw a regular pentagon with just a straightedge? (I think you may also call it a ruler.) If the answer is "yes", then how?

No compass is allowed!

Moreover, the construction must be absolutely precise.

It seems that it's quite a popular problem. I first read through it on a particular site; the suggested answers which some users provided were unfortunately incorrect (it's in Russian though, but I can provide the link to my problem if anyone wants). I tried googling, but I found no answer. Multiple searches, Wikipedia, Youtube—all in vain.

Alexander
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  • If one circle and its center are given, see Poncelet-Steiner theorem – NDB Aug 08 '23 at 15:56
  • "it's in Russian though, but I can provide the link to my problem if anyone wants" Yes, you should edit that into your question. That said, it seems impossible with just a straightedge, unless you really do mean a ruler (in which case, I have no idea). – Xander Henderson Aug 08 '23 at 16:02
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    If nothing is given, then no. For any construction with straightedge (imagine it draw on transparent glass), project it to another plane slanted with respect to the glass. The projection would also satisfy the same (description of the) construction, but if there was a regular pentagon before, now it is not regular. So, the construction could not have guaranteed that the pentagon is regular. Other examples of what could be given are: An arc of a circle and its center. Two concentric circles. Three circles. ... – NDB Aug 08 '23 at 18:38

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No. The construction of a regular pentagon entails the construction of a line segment of length $\sqrt{5}/2$. This is a big problem for your scenario in the following way: the base field of constructible numbers is $\mathbb{Q}$, which is closed under the addition/subtraction and multiplication/division; here, elements of $\mathbb{Q}$ are represented by line segments of rational length and the results of the aforementioned operations by their intersections. That is, if you're working purely with straightedge, then you cannot construct any non-rational lengths. But of course $\sqrt{5}/2$ is known to be irrational; thus you cannot construct it without using circles, which means that a regular pentagon cannot be constructed without a compass.

H. sapiens rex
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