Proposition I-22

Proposition I-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 an manner should be greater than the remaining one.
Proof.

Construct a segment of length of B with extremity at G.

Construct a circle of radius of B with extremity at G.

The three lines are A, B, and C.

We assume that the sum of B and C is greater than A.

Construct a line segment DE with length greater than A.

Construct a line segment FG equal to A on DE.

Construct a segment of length of B with extremity at F.

Construct a circle of radius the length of B with center F.
(Post-3)

At the intersection of the circles, H, join the lines FH and GH.

The resulting triangle FGH has the required lengths, as was to be proved.

QED

(CN-x)

(Prop I-xx)

(Post-xx)