Proposition I.3

Construct equal segments by section

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. I.post.3

Now, since the point A is the centre of the circle DEF, AE is equal to AD. I.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. I.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.

References

graph BT I_3[ I.3 ]:::prop; click I_3 "/elem.1.3" "Book I Proposition 3"; %%%% dependencies I_2[ I.2 ]:::prop; click I_2 "/elem.1.2" "Book I Proposition 2"; I_post3( I.ax.3 ):::post; click I_post3 "/elem.1.post.3" "Book I Postulate 3"; I_def15( I.def.15 ):::def; click I_def15 "/elem.1.def.15" "Book I Definition 15"; I_cn1( I.cn.1 ):::cn; click I_cn1 "/elem.1.c.n.1" "Book I Common Notion 1"; %%%% links I_3 --> I_2 I_3 --> I_post3 I_3 --> I_def15 I_3 --> I_cn1 %%%% I_2 dependencies I_1[ I.1 ]:::prop; click I_1 "/elem.1.1" "Book I Proposition 1"; I_post1( I.post.1 ):::post; click I_post1 "/elem.1.post.1" "Book I Postulate 1"; I_post2( I.post.2 ):::post; click I_post2 "/elem.1.post.2" "Book I Postulate 2"; I_post3( I.post.3 ):::post; click I_post3 "/elem.1.post.3" "Book I Postulate 3"; I_def15( I.def.15 ):::def; click I_def15 "/elem.1.def.15" "Book I Definition 15"; %%%% I_2 links I_2 --> I_1 I_2 --> I_post1 I_2 --> I_post2 I_2 --> I_post3 I_2 --> I_def15 %%%% I_1 dependencies I_post1( I.post.1 ):::post; click I_post1 "/elem.1.post.1" "Book I Postulate 1"; I_post3( I.post.3 ):::post click I_post3 "/elem.1.post.3" "Book I Postulate 3"; I_def15( I.def.15 ):::def click I_def15 "/elem.1.def.15" "Book I Definition 15"; I_cn1( I.c.n.1 ):::cn click I_cn1 "/elem.1.c.n.1" "Book I Common Notion 1"; %%%% I_1 links I_1 --> I_post1 I_1 --> I_post3 I_1 --> I_def15 I_1 --> I_cn1