354 -Dr. R. Key's Propositions containing 
one base; a right line indefinite in length: and one base has 
one vertex. 
Thus : if GH and any other chords pass through the vertex 
V ; the tangents from their extremities meet respectively in 
points of one indefinite line LD : and, conversely, if tangents 
be drawn from L and other points of LD ; all their tangent- 
chords pass through one point V. 
Demonstration. Fig. 2. Let E be the centre, and PQ the 
chord bisected in V. Draw the tangents GL, HL, PD, QD; 
and draw DE, DL, LE, EP, EO. EH. On the diameters DE, 
LE, describe circles. That on DE passes through P and Q, 
because of the right angles DPE, DOE ; and that on LE 
through G and H. And DE passes through V, because of the 
similar and equal right-angled triangles DEP, DEQ. By the 
circle on DE, the rectangle DVE is equal to PV^ ; which, by 
the given circle GPO, is equal to GVH ; whence GEHD is* a 
circle, and is that on LE, and LDE is-f a right angle. In like 
manner, if any other chord ST pass through V, and if tangents 
from S and T would meet in R, RDE would be proved a right 
angle. Therefore LDR is one right line. And conversely 
(fig. 1), if L be any point in a base to V; its tangent-chord 
GH passes through V. If not, let another chord G/i so pass. 
Then (as proved) the tangent from /i meets GL in that base, 
therefore in L ; that is, in the same point as the tangent from 
H. Which is impossible. Therefore GH passes through V. 
S E.D. 
Cor. 1. Fig. 1. The indefinite base to a given vertex V, 
is a line parallel to the chord bisected in V, and passing 
through any point, L or D, in whose tangent-chord V is. 
* By a necessary converse of III End. prop. 35. + Hi Euc). prop. 31. 
