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.
Let the three given straight lines
be A, B,
and of these let two taken together in any manner be
greater than the remaining one,
A, B greater than C,thus it is required to construct a triangle out of straight lines equal to A, B, C.
A, C greater than B,
B, C greater than A;
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.
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, FG 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
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 fo the circle LKH,
GH is equal
But GH is equal to C;
therefore GK is 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 constucted. Q.E.F.
Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)
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