Proposition 2: “From a given point, to draw a straight line equal to a given finite straight line”. In other words, given a point A and another finite straight line CB, draw a line equal to CB at A.



Proof.
Let the given point be A and the given finite line be CB.

Use postule 1 to draw a line between A and one of the end points say B.

Next, use proposition 1 to draw an equilateral triangle from the drawn line AB.

Next, describe a circle with center B and radius CB using postulate 3

Extend the line DB all the way till intersects the circle at E using postulate 2.

Now describe another circle with center D and radius DE (postulate 3).

Extend the line DA until it intersects the circle at F.

The claim is that AF=BC which is what we want. To see this, we know that DF=DE by definition 15. We know that DA=DB by construction. Therefore, BE=AF by common notion 3. But CB=BE by definition 15 and so CB=AF (common notion 3) as required.



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