Book I, Proposition 3

To cut off from the greater of two given unequal straight lines a straight line equal to the less.
# Statement Reason
1. Straight line exists. given
2. Straight line , greater than , exists. given
QEF. Given the two straight lines AB and C, AE has been cut off from AB the greater equal to C the less.

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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.
Construct an equilateral triangle on a given finite straight line.
Place a straight line equal to a given straight line with one end at a given point.