S6^ Dr, R. Key's Propositions containing 
whence (by the present proposition) the tangent-chord of each 
of the tvvo points will pass through the other. Q. E, D. 
Cor. If either LMR be a right angle, or the squares of LG 
and RS be together equal to that of LR ; the other of these 
two follows from it.* 
Prop. V. Fig. 5 . If a diameter be cut by a perpendicular 
from any point of the circumference, and, produced, be cut by 
a tangent from the same point, and if an angle be made at any 
point of the circumference, by right lines to the two intersec- 
tions; this angle and its supplement are bisected by right lines 
to the extremities of the diameter. 
Thus : if LR, a diameter of GNH, be cut in I by NI, and, 
produced, in K by NK, and if GK cut the circumference in H 
between G and K ; the angles INK, IKK, IGK, are bisected 
by NR, HR, GR, and their supplements INX, IHY, IGY, by 
NL, HL, GL ; KN being produced to X, and KG to Y. 
Dem, Produce NI to O in the circumference, and draw OR, 
OL. Then RNK is “f equal to NOR, so to ONR or INR ; also 
LNX to NOL, so to ONL or INL. Produce GI, HI, to T, S, 
in the circumference. Draw HT, SK, SR, SG, SL. Then, 
since HS and GT bisect NO, HKR or GKR is X equal to RKS 
or RKT, HK § to TK, GK to SK ; and HT and GS are per- 
pendicular to LR. IHR or SHR is equal to SGR, so to GSR, 
which, having the supplement || RHG, is equal to RHK. And, 
from the equal arcs RH, RT, TGR or IGR is equal to RGH 
or RGK. And the right lin.es LG, LS, are equal, and the arcs 
LG, LS; whence LHS or LHI is equal to LHG or LHY. 
♦ Prop. III. and IV. 
III Eucl. prop. 8. 
f III Eucl. prop. 32. 
11 III Eucl. prop. 22. 
t Cor. 3 to prop. I. 
