Book I · Proposition 1

Construct an equilateral triangle

“To construct an equilateral triangle on a given finite straight line.”

Visualizing the goal

Begin with the given segment and construct an on it. The finished triangle must have three equal sides: = = .

Follow the construction.

1
Step 1
Let be the given finite straight line. It is required to construct an on the straight line .
2
Step 2
Describe the with center and radius .
3
Step 3
Again describe the with center and radius .
4
Step 4
Join the straight lines and from the point at which the circles cut one another to the points and .
5
Step 5
Now, since the point is the center of the circle , therefore equals .
6
Step 6
Again, since the point is the center of the circle , therefore equals .
7
Step 7
But was proved equal to , therefore each of the straight lines and equals . And things which equal the same thing also equal one another, therefore also equals . Therefore the three straight lines , , and equal one another.
8
Step 8
Therefore the triangle is , and it has been constructed on the given finite straight line . Q.E.F.