Is this a valid demonstration of how to draw a line parallel to another?

[ZeHgS]

Hi! I'm studying Euclid's Elements and I was trying to demonstrate on my own how to draw a line CD parallel to a line AB from a given point E outside of AB.

Is this valid or have I made any wrong assumptions? Here is the drawing:



The steps I took were the following:

1. Connect point E, which will be the origin of the parallel line, to point A.

2. Using Proposition 11, draw a line 's' perpendicular to line AB going through point A.

3. Using Proposition 3, cut off IA = EA from line 's'. This forms the triangle EAI.

4. Using Proposition 2, place GA = EA on the other side. (I think this is where I might have made a mistaken assumption. Does Proposition 2 allow me to transfer a line along with the inclination?).

5. Connect F to G.

6. Now that I have a parallel line as the basis, I can extend it indefinitely as needed then use Proposition 3 to cut off CD from it, which will be parallel to AB and will have the point E as its origin. QEF.

 

[Ishuda]

Hi! I'm studying Euclid's Elements and I was trying to demonstrate on my own how to draw a line CD parallel to a line AB from a given point E outside of AB.

Is this valid or have I made any wrong assumptions? Here is the drawing:



The steps I took were the following:

1. Connect point E, which will be the origin of the parallel line, to point A.

2. Using Proposition 11, draw a line 's' perpendicular to line AB going through point A.

3. Using Proposition 3, cut off IA = EA from line 's'. This forms the triangle EAI.

4. Using Proposition 2, place GA = EA on the other side. (I think this is where I might have made a mistaken assumption. Does Proposition 2 allow me to transfer a line along with the inclination?).

5. Connect F to G.

6. Now that I have a parallel line as the basis, I can extend it indefinitely as needed then use Proposition 3 to cut off CD from it, which will be parallel to AB and will have the point E as its origin. QEF.

First of all, I believe you are 'talking' as though F and E are the same point. Is this true?

If it is true, what you have done can be used to produce the parallel line IMO. Whether it would be easier or not I would approach the problem by extending line AB, draw a perpendicular to AB through E, then draw a perpendicular through E.

 

[ZeHgS]

First of all, I believe you are 'talking' as though F and E are the same point. Is this true?

If it is true, what you have done can be used to produce the parallel line IMO. Whether it would be easier or not I would approach the problem by extending line AB, draw a perpendicular to AB through E, then draw a perpendicular through E.

Thanks for the reply!

Yes, they are, I don't know why I didn't use E lol. But how do you draw a perpendicular through E? I couldn't figure that out. The way I used proposition 11 in my original post isn't the same thing, I extend AB to the left through point A, then cut off a length equal to AB to the left. Then I construct equilateral triangles on both sides and connect their tips to create the perpendicular through A. I can't do this to E because, unlike point A, it's not on the line itself, and I couldn't figure out how to find the corresponding point. How is it done?

 

[ZeHgS]

Lol never mind the very next proposition was actually about how to draw a line perpendicular to another from a given point not on it. I didn't see because I'm taking this very slowly, and I always pause after the propositions to try and think of some myself. I really liked Euclid's trick of taking an arbitrary point on the other side of the line to create a radius.