Proposition 22.
Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one. [
I. 20]
Let the three given straight lines be
A,
B,
C, and of these let two taken together in any manner be greater than the remaining one, namely
A,
B greater than
C,
A, C greater than B, and
B,
C greater than
A; thus it is required to construct a triangle out of straight lines equal to
A,
B,
C.
Let there be set out a straight line
DE, terminated at
D but of infinite length in the direction of
E, and let
DF be made equal to
A,
FG equal to
B, and
GH equal to
C. [
I. 3]
With centre
F and distance
FD let the circle
DKL be described; again, with centre
G and distance
GH let the circle
KLH be described; and let
KF,
KG be joined;
I say that the triangle
KFG has been constructed out of three straight lines equal to
A,
B,
C.
For, since the point
F is the centre of the circle
DKL,
FD is equal to FK.
But
FD is equal to
A;
therefore KF is also equal to A.
Again, since the point
G is the centre of the circle
LKH,
GH is equal to GK.
But
GH is equal to
C;
therefore KG is also equal to C. And FG is also equal to B; therefore the three straight lines
KF,
FG,
GK are equal to the three straight lines
A,
B,
C.
Therefore out of the three straight lines
KF,
FG,
GK, which are equal to the three given straight lines
A,
B,
C, the triangle
KFG has been constructed.
Q. E. F.
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