Book I, Proposition 1

To construct an equilateral triangle on a given finite straight line.
# Statement Reason
1. Straight line exists. given
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QEF. The triangle ABC is equilateral, and it has been constructed on the given finite straight line AB.

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Add to proof Summary
Definition of a point.
Definition of a straight line => A line exists
Definition of a straight line => Choose an arbitrary point
Definition of a circle
Definition of an equilateral triangle.
Draw a straight line from any point to any point.
Extend a finite straight line continuously in a straight line.
Draw a circle with center and radius.
All right angles equal one another.
The Parallel Postulate
Things equal to the same thing are equal to each other.
If equals are added to equals, then the wholes are equal.
If equals are subtracted from equals, then the remainders are equal.
Things which coincide with one another equal one another.
The whole is greater than the part.