THE comes, BOOKS VI, YU 
169 
The same is true if A A' is the minor axis of an ellipse and p 
the corresponding parameter (VII. 2, 3). 
If A A' be divided at H' as well as H (internally for the 
hyperbola and externally for the ellipse) so that H is adjacent 
to A and H' to A', and if A'H: AH = AH': A'H' = A A': p, 
the lines AH, A'H' (corresponding to p in the proportion) are 
called by Apollonius homologues, and he makes considerable 
use of the auxiliary points H, H' in later propositions from 
VII. 6 onwards. Meantime he proves two more propositions, 
which, like VII. 1-3, are by way of lemmas. First, if CD be 
the semi-diameter parallel to the tangent at P to a central 
conic, and if the tangent meet the axis A A' in T, then 
PT 2 : CD* = NT : GN. (VII. 4.) 
Draw AE, TF at right angles to G A to meet CP, and let AE 
meet PT in 0. Then, if p' be the parameter of the ordinates 
to CP, we have 
Ip' : PT = OP : PE (I. 49, 50.) 
= PT:PF, 
or \p' .PF = PT\ 
PT 2 :CD* = i p. PF:\p'.CP 
= PF.GP 
Therefore