Return to Vignettes of Ancient Mathematics
Return to Elements I, introduction
Go to prop. 2

Note Blue text is not part of Euclid's text. Square brackets indicate that the text may not have originally been part of Euclid's text.
(plain diagram)

(general diagram)
Statement
Prop. 1: On the given finite straight-line to construct an equilateral triangle.

Display
(diagram 1) Let the given finite straight line be AB.

Determination
It is, in fact, required to construct on straight-line AB an equilateral triangle.

Construction
(diagram 2)
With a center, A, and a distance, AB, let a circle be described, BGD, (diagram 3)
and again with a center, B, and a distance, BA, let a circle be described, AGE, (diagram 4)
and from point G, at which the circles cut one another, to points A, B, (diagram 5)
let straight-lines be joined, GA, GB. 

Demonstration
(diagram 6) or use (general diagram)
And since point A is center of circle GDB, AG is equal to AB. Again since point B is center of circle GAE, BG is equal to BA. But GA was also shown equal to AB. Therefore, each of GA, GB is equal to AB. But things equal to the same are also equal to one another. Therefore, GA is also equal to GB. Therefore, the three, GA, AB, BG, are equal to one another.

Conclusion
Therefore, triangle ABG is equilateral and has been constructed on the given finite straight-line, AB[. therefore, on the given finite straight-line an equilateral triangle is constructed], just what it was required to make.