LOCUS. 



recefTarily falls in'the circumference of ihe femicircle AD E ; 

 confequeiitly A C, C B and C U, are all equal to each other. 

 But A C, being half of A B, is given, therefore D C is alfo 

 given, whence the locus of the point of bifeftion C is a 

 circle dcfcribed from D with the radius D C. 



Comp'i/ilian — ^From D, witli a dillance equal to half the 

 given line, dcfcribe a circle ; this is the locus required. 



For draw the radius D C, make AC = DC, and produce 

 A C to B. Becaufe A C = D C, the anple A D C = 

 D A C ; but the angles D A C and D B C are together 

 equal to a right angle, and therefere equal to A D C and 

 B D C ; whence the angle D B 6 is equal to the angle 

 B D C, and, confequently, the fide D C is equal to B C. 

 The fegments A C, B C are thus each of them equal- to 

 i) C, and hence A B is itfelf double D C, or is equal to 

 tlie given ftraight line. 



Pkoi'. V. 



If from two given points there he inflefted two ftraiglit 

 lines in a given unequal ratio, the lo;u8 of their point of 

 concourfe is a given circle. 



Let A C and B C, drawn from the points A, and B, 

 have a given ratio, but not that of equality ; then will C, 

 the point of concourfe, lie in the circumference of a given 

 circle. Fi^. 5. 



yliiiilyjis. — Draw C D, making the angle BCD equal to 

 B A C ; and meeting A B produced in D. The triangles 

 J) A C and D C B, having the angle at D common, and 

 the angles at A and C equal, are evidently fimilar ; and 

 hence A D : A C : : C D : C B, and alternately, A D : 

 C D : : A C : C B, that is, in the given ratio ; but A D : 

 C D : : C D : B D, and confequently, A D is to B D in 

 the duplicate of the given ratio A D to CD, and which is 

 therefore likewife given. Confequently B D, and the point 

 D, are given ; and B D being thence given, its extremity 

 C muft lie in the circumference of a circle defcribed with 

 that radius. 



Compqft'ion. — Divide A B in the given ratio in E, and in 

 the fame ratio make ED to B D ; the circle defcribed 

 from the centre D, and with the radius D E, is the locus 

 required. 



For fince A E : E B : : E D ; B D, it follows that 

 A D : E D : : C D : E D, or as C D : B D ; hence the 

 triangles D A C and D C B, thus having their fides, which 

 contain their common angle D, proportional, are fimilar ; 

 and tiierefore A C : A D : : B C : C D, or alternately, 

 A C : B C : : AD : CD or D E, that is, in the given 

 ratio. 



Prop. VI. 



If two (Iraight lines, containing a given rectangle, be 

 drawn from a given point at a given angle :• (hould the one 

 terminate in a itraight line given in pofition, the other will 

 terminate in the circumference of a given circle. Fig. 6. 



Let the point A, the angle B A C, and the rcdlangle 

 under its fides B A, A C, be given ; if the direction B D 

 "be given, then will the locus of C be a given circle. 



yinalyjis. — From A let fall the perpendicular A D upon 

 B D ; draw A E, to contain with A D an angle equal to 

 the given angle, and a reftangle equal to the given fpace, 

 and join C E. 



Since A D is evidently given in pofition and magnittide, 

 A E is likeuife given in pofition and magnitude ; and the 

 rectangle A D x A E being equal to A B x A C, there- 

 fore A D : A B :: A C : A E ; but the angle D A E is 

 equal to BAC, and hence D A B is equal to E A C. 

 Wlierefore the triangles A B D, A E C, having each an 



equal angle, and the fides containing it proportional, are 

 fimilar ; and confequently the angle A C E is equal to the 

 right angle A D B. Whence the locus C is a circle, having 

 A E for its diameter. 



Compojition. — Having let fall the perpendicular. A D, 

 draw A E, making the angle DAE equal to the given 

 r.ngle, and the reftan^les D A, A E, equal to the given 

 fpace. On A E as a diameter defcribe a circle ; this is the 

 locus required. For join C E, and the triangles DAE, 

 E A C, being right-angled at D and C, and haviijg the 

 vertical angles at A equal, are evidently fimilar ; and confe- 

 quently AD : AB :: AC: AE; and hence the rec- 

 tangle xAB X AC=::ADxAE, that is, it is equal to 

 the given fp3ce. 



The foregoing propofition we have drawn with little va- 

 riation from the chapter on loci given by proit-fiiir Lefiie 

 in his Gi-ometry, and feveral of the following propofitions 

 are hkewife derived from the fame fource. 



7. If a ilraight line drawn from a given point to a (Iraight 

 line given in pofition, contain a given redtangle, the locus 

 of Its point of feiHion will be a given circle. 



8. If two Itraight lines in a given ratio, and containing a 

 given angle, terminate in two diverging lines, wliich are 

 given in pofition, the locus of their vertex will likewife be 

 a right line given in pofition. 



9. If from two points there be drawn two {Iraight lines, 

 of whofe fquares the difference is given, the locus of their 

 point of concourfe will be a right line given in pofition : 

 or, which is the fame, if the bafe of a triangle, and the 

 difference of the fquares of the two fides be given, the 

 vertex of the triangle will fall in a right line given in 

 pofition. 



10. "If the bafe and vertical an^le of a triangle be given, 

 the locus of its vertex will be the circumference of a given 

 circle. 



11. If the difference of the fides, and the radius of the 

 infcribed circle of a triangle be given, the locus of its ver- 

 tex is a right line given in pofition. 



12. If two given unequal perpendiculars be drawn to a 

 right line given in pofition, and their oppofite extremities 

 be joined, the locus of the point of interfettion will be a 

 right hnc given in polition. 



13. If in any triangle the bafe be given, and tiie fnm of 

 the fquares of the other two fides, the locus of the vertex is 

 a given circumference. 



14. If from given points there be drawn (Iraight lines, 

 whole fquares are together equal to a given fpace, their 

 point of concourfe will terminate in the circumlcrence of a 

 given circle. 



i^. If right lines be drawn from a given point to cut a 

 given circle, and from the points of intcrfeflion there be 

 taken, upon thefe lines, on either fide, lines in a conftant 

 given ratio to tlie diftance between the refpeftive points of 

 interfeftion and the given point ; the locus of the points, fo 

 determined, will be a circle. 



16. If two circles cut each other, and through cither 

 point of interfeflion a right line be drawn, cutting both 

 the circles, then, if a right line be always taken thereon 

 from one of thofe points in a given ratio to the part inter- 

 cepted between the circles, the locus of the points fo deter- 

 mined will be a circle. 



17. If the circles cut each other as above, and a right 

 line be drawn through either interfeclion, cutting both the 

 circles, then if a right line be always taken thereon from 

 one of thofe points in a given ratio to the part between the 

 other point and interfe£tion, the locus of the point fo deter- 

 mined will be a circle. 



18. If 



