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How can I construct a square using only a pencil and a compass, i.e. no ruler.

Given: a sheet of paper with $2$ points marked on it, a pencil and a compass.

Aim: plot $2$ vertices such that the $4$ of them form a square using only a compass.

P.S.: no cheap tricks involved.

E. Joseph
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TANUJ
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3 Answers3

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On way to do this is to start by constructing the middle point of your segment like this:

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Then you will easily have the middle of your square like this:

enter image description here

And you know can have a circonscrit circle for your square, and constructing the square is finally possible!

Edit.

Since you asked for it, I have made a few more drawings to illustrate how to construct the point $O$ from where we left it.

enter image description here enter image description here enter image description here enter image description here enter image description here

Where the last circle has for radius $CF$ and for center $A$.

Edit 2.

Since more details were requested, here is how to finish the proof once $O$ has been constructed.

enter image description here enter image description here enter image description here enter image description here enter image description here

E. Joseph
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  • How do you construct point "O" in the last figure? – Intelligenti pauca Feb 05 '17 at 14:36
  • @Aretino I edited to specify that particular point. – E. Joseph Feb 05 '17 at 15:11
  • But I still don't understand how would you join the points ? –  Feb 05 '17 at 16:20
  • @A---B You don't need to. When you have $O$, you can draw a circle that will pass trough every angle of your square, and it is now easy to construct the square. – E. Joseph Feb 05 '17 at 16:23
  • I understand your construction but we still need to join the vertices of square on the circumference of the circle ? –  Feb 05 '17 at 16:25
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    @A---B I did more drawings, and I hope it answers what you are asking me. – E. Joseph Feb 05 '17 at 16:39
  • your answer is certainly correct,but if you had noted that that we can consrtuct a square of side BC trivially with vertex O.So what if we do that right from the beginning.i.e given a side AB,we can construct a square with side AB/2.so lets just double AB – TANUJ Feb 07 '17 at 16:50
  • @TANUJ I think you are right, my answer is certainly not the fastest one! – E. Joseph Feb 07 '17 at 22:20
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The key to solve this problem is how to construct $\sqrt{2}$. enter image description here

Seyed
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The only difficulty is to create Perpendicular lines. One possible way could be, if you are precise enough:

lets suppose the vertices are (P1,P2,P3,P4), where P1 and P2 are given two points.

Join the two points(P1,P2) and extend on both sides.

To draw a perpendicular at point(P1) on the line: Mark two points on the line equidistant from P1 on both sides, then place your compass at the marked points, there after create arcs using your compass such that they intersect.

joining the intersection of arcs and P1 would give you a perpendicular line on the given line.

Since you have created a perpendicular line, then it's not difficult to create a Square from it.

Pushkar Soni
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