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 toAD
; so thatAE
is also equal toC
. 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