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Euclid's Elements: Book I: Proposition 3

Proposition 3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.

Let AB, C be the-two given unequal straight lines, and let AB be the greater of them.

Thus it is required to cut off from AB the greater a straight line equal to C the less.

At the point A let AD be placed equal to the straight line C; I. 2
and with centre A and distance AD let the circle DEF be described. Post. 3

Now, since the point A is the centre of the circle DEF, AE is equal to AD. Def. 15
But C is also equal to AD.
Therefore each of the straight lines AE, C is equal to AD;
so that AE is also equal to C. C.N. 1

Therefore, given the two straight lines AB, C, from AB the greater AE has been cut off equal to C the less.

(Being) what it was required to do.

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