Proposition 22
Construction
To construct a triangle out of three line segments which equal three given line segments, the sum of every two of which is greater than the third.
Construction Steps
1. Let AB , CD and EF be the given line segments, of which the sum of every two is greater than the third.
2. Draw a lineGH such that ￜGH ￜ = ￜAB ￜ .
3. Construct the circle centered at G with radiusￜCD ￜ and the circle centered at H with radius ￜEF ￜ . These two circles intersect at two points. Pick one intersection and name it P.
4. JoinPG and PH . △PGH is constructed with ￜGH ￜ = ￜAB ￜ , ￜGP ￜ = ￜCD ￜ and ￜHP ￜ = ￜEF ￜ .
2. Draw a line
3. Construct the circle centered at G with radius
4. Join
Original statement
ἐκ τριῶν ϵὐθϵιῶν, αἵ ϵἰσιν ἴσαι τρισὶ ταῖς δοθϵίσαις ϵὐθϵίαις, τρίγωνον συστήσασθαι: δϵῖ δὲ τὰς δύο τῆς λοιπῆς μϵίζονας ϵἶναι πάντῃ μϵταλαμβανομένας διὰ τὸ καὶ παντὸς τριγώνου τὰς δύο πλϵυρὰς τῆς λοιπῆς μϵίζονας ϵἶναι πάντῃ μϵταλαμβανομένας.
English translation
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.
