One of the oldest and most influential mathematical treatises of all time is the Elements, a series of thirteen books by the ancient Greek mathematician Euclid of Alexandria. The constructions described in the Elements can be represented in the Wolfram Language using GeometricScene and visualized with RandomInstance.
Proposition 1 of Book I states that given any two points and , one can construct an equilateral triangle having and as two of its vertices. In particular, draw two circles centered at and , respectively, whose radii are equal to the distance between them. Then their point of intersection forms the third vertex of such an equilateral triangle.
Proposition 22 of Book I generalizes Proposition 1 by stating that for any positive quantities , and , such that , there is a triangle having side lengths , and .
Randomly choose positive quantities , and , such that .
The construction proceeds as follows: construct a straight line through the points , , and in order, with and distance apart, and distance apart and and distance apart. Draw the circle centered at going through , as well as the circle centered at going through . If is one of the points where these circles intersect, then is distance from , is distance from and is distance from . Thus the points , and form such a triangle.