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In Euclid’s Elements Book I, Proposition 31 constructs a line parallel to a given line through a given point. In the proof of Proposition I.31, Euclid appeals to Proposition I.23; in the proof of Proposition I.23 he uses Proposition I.22; and in I.22 he uses the construction from I.3. All of these are geometric constructions — they involve drawing circles and intersection points — but most presentations gloss over the intermediate steps.

Most videos and explanations I’ve found show what needs to be constructed but then skip the detailed classical steps: they freehand the final segments without showing the actual circle constructions or the intermediate intersections needed to justify the parallels.

What I’m looking for:

A video, animation, or resource that literally draws out every step of the Euclidean straightedge-and-compass constructions used in Proposition I.31, including the auxiliary constructions from I.23, I.22, and I.3.

Alternatively, a step-by-step geometric derivation — with diagrams — that shows all circles, intersections, and straightedge moves needed to produce the final parallel line from a given line and point.

Is there a resource like this, or could someone provide an embedded, step-by-step construction with all the intermediate circles and lines shown?

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  • $\begingroup$ Why don't you just screenshot each step in Geogebra? $\endgroup$ Commented Feb 27 at 16:47
  • $\begingroup$ @RobinSparrow I don't know how to do all the constructions in I.31 myself. That is why I am outsourcing it to someone who knows how to do it. If I knew how to do the proof with all the constructions, I would've done that already. $\endgroup$ Commented Feb 27 at 16:59
  • $\begingroup$ aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html gives a standard translation of Elements. At what point, exactly, are you losing the thread of what to do? All you need to do is build an arbitrary triangle on a line, then make a copy of that triangle on one of its sides, extended. $\endgroup$ Commented Feb 27 at 17:16
  • $\begingroup$ @RobinSparrow The orientation of angles DAE and ADC is throwing me off, and I'm lost immediately after drawing DA $\endgroup$ Commented Feb 27 at 19:25
  • $\begingroup$ The alternate interior angles are labeled in a way that respects the symmetry between them. Imagine putting a pin in the midpoint of segment $DA$ and rotating the diagram half way around $(180^\circ)$. Notice that $\angle DAE$ would land on $\angle ADC$ with labels in that order/orientation. $\endgroup$ Commented Feb 27 at 19:52

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On line $AB$ construct an arbitrary triangle $BCD$, take a side extended, say $BC$ and at some arbitrary point $B'$ draw circles of radius $BC$ and $BD$ (red and green), where the circle of radius $BC$ intersects $BC$ label $C'$, then from $C'$ draw the circle of radius $CD$ (yellow) and where this circle intersects the (green) circle of radius $BD$ label $D'$. Then by SSS $\triangle{BCD} \cong \triangle{B'C'D'}$ hence at the traversal $BB'$ the lines $BD$ and $B'D'$ have equal alternate angles and are therefore parallel.

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