164 On the ORIGIN and 
DH, that is, to the fquares of DL and LH, the rectangle 
K HK’ is equal to the fquare of HB, fo that HB touches the 
circle BK K’,. But BG is at right-angles to HB ; therefore the 
centre of the circle BK K’ is in the line B G3; and it is alfo in 
the line D E; therefore G is the centre of the circle BK KY, and 
GB is equal a GK. 
Tuus we have an inftance of a problem, and that too 
a very fimple one, which is in general determinate, admitting 
only of one folution, but which neverthelefs, in one particular 
cafe, where a certain relation takes place among the things 
given, becomes indefinite, and admits of innumerable folu- 
tions. The propofition which refults from this cafe of 
the problem is a Porifm, according to the remarks that were 
made above, and in effect will be found to coincide with the 
66th propofition in Dr Srmson’s Reftoration. It may be thus 
enunciated: “ A circle ABC being given in pofition, and alfo a 
ftraight line D E, which does not cut the circle, a point K may 
be found, fuch that if G be any point whatever in the line DE, 
the ftraight line drawn from G to the pdint K, fhall be equal to 
the ftraight line drawn from G, touching the circle A BC.” 
ro. Tue following Porifm is alfo derived in the fame man- 
ner from the folution of a very fimple problem : 
PROP. If) PR OB hece. 
A TRIANGLE ABC being given, and alfo a point D, to 
draw through D a ftraight line DG, fuch, that, perpendi- 
culars being drawn to it from the three angles of the tri- . 
angle, viz. AE, BG, CF, the fum of the two perpendi- 
culars on the fame fide of DG, fhall be equal to the re-- 
maining 
