!6 2 On the ORIGIN and 



PROP. L P R O B. Fig. i. 



A circle A B C, a ftraight line D E, and a point F, being 

 given in pofition, to find a point G in the ftraight line 

 D E, fuch that G F, the line drawn from it to the given 

 point, fhall be equal to G B, the line drawn from it touch- 

 ing the given circle. 



Suppose the point G to be found, and GB to be drawn 

 touching the circle ABC in B ; let H be the centre of the 

 circle ABC; join H B, and let H D be perpendicular to D E ; 

 from D draw D L, touching the circle A B C in L, and 

 join H L. Alfo from the centre G, with the diftance G B or 

 G F, defcribe the circle B K F, meeting H D in the points K 

 and K'. 



It is plain, that the lines H D and D L are given in pofition 



and in magnitude. Alfo, becaufe G B touches the circle A B C, 



H B G is a right-angle ; and finee G is the centre of the circle 



B K F, therefore H B touches the circle B K F, and confequent- 



ly the fquare of H B, or of H L, is equal to the rectangle 



K' H K. But the rectangle K.' H K, together with the fquare 



of D K, is equal to the fquare of DH, becaufe K K' is bifected 



in D ; therefore the fquares of H L and D K. are alfo equal to 



the fquare of D H. But the fquares of H L and L D are equal 



to the fame fquare of DHj wherefore the fquare of DK is e- 



qual to the fquare of DL, and the line D K to the line DL 



But D L is given in magnitude ; therefore D K is given in 



magnitude, and K is therefore a given point. For the fame 



reafon, K' is a given point, and the point F being alfo given by 



hypothecs, the circle B K F is given in pofition. The point G 



therefore, the centre of the circle BK F is given, which was to 



be found. 



Hence 



