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Euclid: Elementa
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Proposition 22. 
PROBL. 8. PROPOS. 22. 
第二十二題 
Out of three straight lines, which are equal to three given straight lines, to construct a triangle: 
EX tribus rectis lineis, quæ sint tribus datis rectis lineis æquales, triangulum constituere.  
三直線。求作三角形。 
thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one. [I. 20] 
Oportet autem duas reliqua esse maiores omnifariam sumptas: quoniam vniuscuiusque trianguli duo latera omnifariam sumpta reliquo sunt maiora. 
其每兩線幷。大於一線也。 
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. 
TRES lineæ rectæ datæ sint A, B, & C, quarum quælibet duæ reliqua sint maiores, (Alias ex ipsis non posset constitui triangulum, ut constat ex propositio 20. in qua ostensum fuit, duo quæuis latera trianguli reliquo esse maiora.)    oporteatque construere triangulum habens tria latera tribus datis lineis æqualia. 
.  .  . 
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. 
Ex assumpta recta quauis D E, infinitæ magnitudinis abscindatur recta D F, æqualis rectæ A.  Et ex reliqua F E, recta F G, æqualis rectæ B; & ex reliqua G E, recta G H, æqualis rectæ C.1   Deinde centro F, interuallo vero F D, circulus describatur D I K.  Item centro G, interuallo autem G H, alius circulus describatur HIK, qui necessario priorem secabit in punctis I, & K, (cum enim duæ F D, G H, maiores ponantur recta F G; Si ex F E, sumatur recta F L, æqualis ipsi F D: & ex G D, recta G M, æqualis ipsi G H, cadet punctum M, inter L, & D. Si namque M, caderet in L, punctum, essent G L, F L, hoc est, G H, & F D, æquales rectæ F G: Si vero M, caderet inter G, & L, essent eædem duæ F L, G M, hoc est, D F, G H, minores recta F G, quorum utrumque est contra hypothesin. Id quod ex appositis figuris apparet) ex quorum quolibet, nimirum ex K, ducantur ad puncta, F, G, rectæ K F, K G,  factumque erit triangulum F G K, cuius latera dico æqualia esse datis rectis 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. 
Cum enim recta F K, æqualis sit rectæ F D,  & recta A, per constructionem eidem F D, æqualis;  erit latus F K, rectæ A, æquale.  Rursus quia G K, æqualis est ipsi G H,  & recta C, eidem G H;  erit quoque latus G K, rectæ C, æquale:  Positum autem fuit per constructionem, reliquum latus F G, reliquæ rectæ B, æquale.  Omnia igitur tria latera F K, F G, G K, tribus datis rectis A, B, C, æqualia sunt. 
.  .  .  .  .  .  .  . 
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. 
Constituimus ergo ex tribus rectis lineis, quæ sunt tribus datis rectis lineis æquales, triangulum:  Quod faciendum erat.

PRAXIS
SVMATVR recta D E, æqualis cuicunque rectarum datarum, nempe ipsi B, quam nunc volumus esse basin. Deinde ex D, ad interuallum rectæ A, arcus describatur: Item ex E, ad interuallum rectæ C, alter arcus secans priorem in F. Si igitur ducantur rectæ D F, E F, factum erit triangulum habens tria latera æqualia tribus datis lineis. Erit enim latus D F, æquale rectæ A, propter interuallum ipsius A, assumptum: & latus E F, ipsi C, propter assumptum interuallum C: D E, vero latus, acceptum est rectæ B, æquale, ab initio.

SCHOLION
HAC arte cuicunque triangulo proposito alterum prorsus æquale & quoad latera, angulosque & quoad aream ipsius, constituemus. Sit namque triangulum quodcunque A B C, cui æquale omni ex parte est construendum. Intelligo eius latera, tanquam tres lineas rectas datas A B, B C, C A, quarum quælibet duæ maiores sunt reliqua. Deinde sumo rectam D E, æqualem uni lateri, nempe B C; & ex D, interuallo lateris A B, arcum describo, item alium ex E, interuallo reliqui lateris C A, qui priorem secet in F, &c. 
.  . 
 
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