Th Visul Construtions of Euli Constrution #11 73 Constrution 11: Book I, Proposition 42 To onstrut, in givn rtilinl ngl, prlllogrm qul to givn tringl. Not: Equl hr mns qul in r.
74 Constrution # 11 Th Visul Construtions of Euli I.42:3. Lt BC th givn tringl, n D th givn rtilinl ngl; D B C I.42:8. Lt BC ist t E, ([I.10]) WNTED Rll. GOSUB I.10.4 B C
Th Visul Construtions of Euli Constrution #11 75 Lt th quiltrl tringl onstrut on it, [I.1] WNTED GOSUB I.1. I.1:7. With ntr n istn lt th irl sri; [Post. 3]
76 Constrution # 11 Th Visul Construtions of Euli I.1:10. gin, with ntr n istn lt th irl sri; [Post. 3] Not: In C#5 (I.10) thr r svrl inssntil stps t this point. Now w omit thm. RETURN to I.10 t lin 4. lt join.
Th Visul Construtions of Euli Constrution #11 77 I.10:8. Th stright lin hs n ist t th point. RETURN to I.42 t lin 9. Rll, lnup. I.42:9. n lt E join, [tully, this lin will not us.] D B E C
78 Constrution # 11 Th Visul Construtions of Euli I.42:10. on th stright lin EC, n t th point E on it, lt th ngl CEF onstrut qul to th ngl D; [I.23] D F WNTED GOSUB I.23 Rll. B E C I.23:8. On th stright lins, rsptivly lt th points, tkn t rnom;
Th Visul Construtions of Euli Constrution #11 79 I.23:10. lt join, I.23:11. (Prphrs) Mov th tringl so th s gos to th lin with th ngl t t th n,. ol s GOSUB I.22P. W follow th summry. Extn th lin.
80 Constrution # 11 Th Visul Construtions of Euli (Mov th rm.) Lot th point so tht is qul to. ([ivirs]) ol s (Mov th s.) Lot th point so tht is qul to. ([Divirs]) ol s
Th Visul Construtions of Euli Constrution #11 81 (Mov th ol rm.) Lot th point i so tht i is qul to. ([Divirs]) ol s i I.22:24. Swing th rm.
82 Constrution # 11 Th Visul Construtions of Euli I.22:26. Swing th ol rm. I.22:28. Join th point whr th two rms mt, f, to th ns of th (mov) s. RETURN to I.23 t lin 11. I.23:18. Th ngl is qul to th ngl f. RETURN to I.42 t lin 10. f
Th Visul Construtions of Euli Constrution #11 83 Extn th si, f. f Clnu. RETURN to I.42 t lin10. I.42:14. Through lt G rwn prlll to EC, [I.31] WNTED D F G GOSUB I.31. W will tk BC s th givn lin, so it is not nssry to rll. W tk th point D of I.31 s B, s not in C#10. D = B is lry join. W tk up I.31 t lin 8. B E C
84 Constrution # 11 Th Visul Construtions of Euli I.31:8. On th stright lin B n t th point on it, lt th ngl onstrut qul to th ngl BC [I.23]; WNTED D F GOSUB I.23 W must mov tringl nlosing th ngl BC, ut BC is lry tringl. Joining I.23 t lin 12, whih is whr I.22 is ll to mov th tringl, ut w must us th Prolus Vrition, I.22P. s usul, w follow th summry, in trms of th s n th two rms, n ol. This tim, w will furthr shortn tht routin y swinging h rm s soon s it is mov. Th intrpolt stp (xtning th trgt lin) is thn unnssry. E B E C I.22P. Mov th rm to th n of th trgt lin B n swing it. (Compss, [Post. 3]) D F s ol B C
Th Visul Construtions of Euli Constrution #11 85 I.22P. Mov th ol rm to th ol n B of th trgt lin B n swing it. D F s ol B ol C I.22P. Connt th mting point E of th n ol irls - th on on th si of th s opposit to th sour tringl - to th n of th trgt lin B. W n not mov th whol tringl BC, s this lin, th mov rm, is th only on n hr. So, RETURN to I.31 t lin 8. Clnup. E F ol B C
86 Constrution # 11 Th Visul Construtions of Euli I.1:11. n lt th stright lin G prou in stright lin with. F G Clnup n RETURN to I.42 t lin 14. B E C I.42:14. n through C lt CG rwn prlll to EF. WNTED D F G GOSUB I.31 Rll. B E C
Th Visul Construtions of Euli Constrution #11 87 I.31:7. Lt point tkn t rnom on, n lt join. I.31:8. On th stright lin, n t th point on it, lt th ngl onstrut qul to th ngl [I.23]; WNTED
88 Constrution # 11 Th Visul Construtions of Euli GOSUB I.23 s w hv lry th lin, w GOSUB I.22P t on, to mov th tringl. Swing th rm roun th n of th trgt lin. ol s Swing th ol rm roun th ol n of th trgt lin. ol
Th Visul Construtions of Euli Constrution #11 89 Connt th mting point, on th si of th trgt lin opposit to th sour tringl, to th n points of th trgt lin. Th tringl hs n mov. RETURN from I.22P to I.23. W hv th ngl qul to th ngl. Rll n RETURN to I.31 t lin 8. Now w xtn th nw lin whih is prlll to, rll, n RETURN to I.42 t lin 14.
90 Constrution # 11 Th Visul Construtions of Euli I.42:29. Thrfor th prlllogrm FECG hs n onstrut qul to th givn tringl BC, in th ngl CEF whih is qul to D. D F G Q.E.F. B E C