This page is interactive and requires JavaScript. The construction tools, mode switching, and proof reconstruction all run in the browser. The non-interactive shape of Proposition I.1 — its statement, postulates, and proof structure — is available in the propositions reference and the foundations page.

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Construct an equilateral triangle on a given finite straight line.

A triangle with three equal sides. You could sketch one in a moment — so why does it stand at the very gate of the most influential book in the history of reason, the proposition every later proof leans back upon? For two thousand years, people who were not fools believed a figure like this touched something the senses cannot reach: that in drawing it exactly, from nothing but permitted moves, you were not inventing a shape but uncovering an order already there. You needn't believe them. But before you decide they were dreaming, build the thing they revered with your own hand, and see what it asks of you.

You will not be told the proof. You will rebuild it — choosing moves that correspond to Euclid's postulates, justifying each inference, and reconstructing the argument once the figure is complete. The tools refuse illegal moves. The goals stay open until you have actually closed the construction.

Free construction · choose a tool

Tool — Straight line Click a point. Then click a second, distinct point, to draw the segment between them.

Goals

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Circle α with centre A and radius AB.Postulate 3
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Circle β with centre B and radius BA.Postulate 3
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A point at the intersection of α and β.Intersection · select
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Segments from that point back to A and to B.Postulate 1

Construction closed

After the reveal

Now reproduce this on paper, by hand, without looking. Compass and straightedge only — no ruler, no measurements. We'll surface the prompt again in 24h.

Justify the proof

First certify that each construction move was licensed. Then demonstrate that the figure Euclid built is actually equilateral — the equality the proposition exists to prove.

What this proof rests on

The minimal foundation Proposition I.1 actually depends on.

Postulate 1To draw a straight line from any point to any point.
Postulate 3To describe a circle with any centre and radius.
Definition 15A circle is a plane figure such that all radii are equal.
Common Notion 1Things equal to the same thing are equal to one another.
Postulate 2Not used here.
Postulate 5Not used here.
→ Browse all foundations

From memory · spaced reconstruction

The figure is hidden. Rebuild the proof in your own words — naming each step and the postulate that licenses it. Your attempts are saved and brought back at increasing intervals; the practice loop closes when you can reproduce the proof days later, not just minutes later.

Practice queue →

Counterexample · find the buried flaw

Construction history

00AB is given. Given

No moves yet. Pick a tool below and act.

0 / 4 goals

Tutor Begin where Euclid begins. The only points you have are A and B; the only segment you have is AB. The first move that makes new geometry available is a circle — but which postulate licenses it, and where does it go?

Commentary

A note on the simplicity of this proof.

Proposition I.1 is the first proposition of Euclid's Elements because it is the smallest non-trivial demonstration of the method: a construction that produces a definite figure, justified from a minimal apparatus, terminating in a claim that follows by logical necessity rather than appearance.

Notice the surprising thinness of what the proof depends on: two postulates, one definition, and one common notion. This thinness is itself part of the lesson — that what looks foundational often is, and what looks decorative often is.

It is also worth noticing what the proof fails to establish, and why the question matters. Euclid does not prove that the two circles must intersect; he assumes it. A complete treatment requires a continuity axiom Euclid did not state. The training value here is partly negative: learning to see the unstated assumption beneath an apparently airtight argument.

When you can rebuild this proof from memory — including the step that establishes AC = BC = AB by reference to Common Notion 1 — you will have done more than learn a proposition. You will have practised the cognitive operation that all of Book I is built upon.