3^0 Dr. R. Hey's Propositions containing 
cut by a chord produced to that point ; the produced chord is 
divided in harmonica! proportion.^ 
Thus ; if NO the tangent-chord of K be cut in V by a chord 
GH, or in I by the diameter LR, each, when produced, pass- 
ing through K ; then GK is to HK as GV to VH, and LK to 
RK as LI to IR. That is ; the distances of G and H, ofL and 
R, from K, are proportional to their distances from V and I 
respectively. 
Pern. Let the figure be as in Prop. V. ; and let GA, HB, be 
perpendiculars on LR. Then the angles HIR, GIL, are*f 
equal. Therefore the triangles HIB, GLA, are similar; and 
GK is to HK as GA to HB, as AI to IB, as GV to VH. Fur- 
ther. LK is to RK as the rectangle LKR or the square of NK 
to the square of RK, as that of J NI to tliat of IR, as § LI to 
IR. Q.E.D.W 
Cor. 1. If a chord GT, through any vertex'!, will, produced, 
cut KC, the indefinite base to I, in an}^ point C ; then GC is 
to TC as GI to IT, and GC to GI as TC to TI. For the tan- 
gent-chord of C passes ^ through I. 
Cor. 2. If GT be a chord through any vertex I, and Z any 
point in the base, and if the chord bisected in I be cut in D by 
ZG, in W by ZT produced ; ID is equal to IW. For DW 
• This proportion has been expressed in different ways. A line LK consists of 
three parts, LI, IR, RK ; and the whole line is to either extreme part, as the other 
is to the middle part. Otherwise: LK, LR, LI, are three lines; and the first is to 
the third, as the difference of the first and second is to that of the second and third. 
f In Cor. 1 to prop. V. % Cor. 2 to prop. V. 
§ VI Eucl. prop. 8. Cor. and prop. 20. Cor. 2. 
IJ Among the proofs which might be given, of this proposition, one (applicable 
both to GH and LR) is by VI Eucl. prop. 3 and prop. A following it, together 
with the Prop. V here given and its first corollary. ^ Prop. I. 
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