162 On the ORIGIN and 
PROP. -l.- PROB. Présts 
Acircte ABC, a ftraight line DE, anda point F, being 
given in pofition, to find a point G in the ftraight line 
DE, fuch that GF, 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 HB, and let HD be perpendicular to DE; 
from D draw DL, touching the circle ABC in L, and 
join HL. Alfo from the centre G, with the diftance GB or 
GF, defcribe the circle BK F, meeting HD in the points K 
and K’. 
Ir is plain, that the lines H D and DL are given in pofition 
and in magnitude. Alfo, becaufe G B touches the circle ABC, 
HBG is aright-angle ; and fince G is the centre of the circle 
BKF, therefore HB touches the circle BK F, and confequent- 
ly the fquare of HB, or of HL, is equal to the rectangle 
K’HK. But the reGangle K’ HK, together with the fquare 
of DK, is equal to the fquare of DH, becaufe K K’ is bifected 
in D; therefore the fquares of HL and DK are alfo equal to 
the fquare of DH. But the fquares of H L and L D are equal 
to the fame fquare of DH; wherefore the {quare of DK is e- 
qual to the fquare of DL, and the line DK to the line DL. 
But DL is given in magnitude; therefore DK is given im 
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 
hypothefis, the circle B K F is given in pofition. The point G 
therefore, the centre of the mee BK F is given, which was to _ 
be found. 
HENcE 
