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I found this problem in a website, but I don't know how to solve it.

Given a line segment $AB$. Construct an equilateral triangle with one side being $AB$.

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The first proposition in the first book of Euclid's Elements. Perhaps a good place to look for self-learners! aleph0.clarku.edu/~djoyce/java/elements/elements.html –  GEdgar May 4 '13 at 13:33
    
The first proposition also contains the first gap. One draws certain circles, not quite the ones mostly used nowadays since the compass is "collapsible." No proof is given that the circles meet, and Euclid's axioms are not sufficient to prove that they do. –  André Nicolas May 4 '13 at 16:14

3 Answers 3

Place the tip of the compass on point A, and open it up so that you can draw an arc across point B.

Your compass radius is now set to the length AB.

Keeping the radius the same, place the tip of the compass on point A again, and draw an arc on the paper above your segment.

Place the tip of the compass on point B now, again using the same radius, an draw another arc that intersects the first arc. The point of intersection is point C.

Now use your straightedge to draw line segments AC and AB.

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With problems like these, I think it is best to just try.

Geogebra is great software that makes it easier to draw geometric constructions. I recently made a little geometry game with geogebra software, and this is actually the first level of that game! You can try that level here.

Hint : Draw circles ! If you heading in the right direction, the software will give an "Well done!" message.

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If a side of your triangle is $[AB]$ then $A$ and $B$ are 2 points out of 3 from your triangle. You want then to find $C$ such that $ABC$ would be equilateral. To do this, just draw the circle of centre $A$ and radius $AB$. Then draw the circle of centre $B$ and radius $AB$. The point of intersection of the circles is the point $C$. You then have your triangle!

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