Step  Example  Description  Justification 
1 

Start with a line segment 

2 

Label the end points A and B. 

3 

Construct a circle with the center at A and the radius the length of segment AB. 
Euclid's Elements Book 1 Postulate 3: A circle can be drawn with any center and any radius. 
4 

Construct a circle with the center at B and the radius the length of segment AB. 
Euclid's Elements Book 1 Postulate 3: A circle can be drawn with any center and any radius. 
5 

Label an intersection of the two circles C. 
6 

Construct a line segment AC and BC. This forms triangle ABC. 
Euclid's Elements Book 1 Postulate 1: A straight line can be drawn from any point to any point. 
7 
Since they are radii of the same circle, the line segment AC
is the same length as the line segment AB. Similarly, the
line segment BC is the same length as the line segment
AB. 
Euclid's Elements Book 1 Definition 15: A circle is all points equidistant from a center point.

8 
Since AB ≡ AC and
AB ≡ BC, then it must be true that
AC ≡ BC. 
Euclid's Elements Book 1 Common Notion 1: If A = B and B = C then A = C.

9 
By the definition of an equilateral triangle, the triangle
ABC is equilateral. 
Euclid's Elements Book 1 Definition 20: An equilateral triangle is a triangle where the length of all three sides is the same.



Q.E.D. 

Table 2 