THE COLLECTION. BOOK VII 
423 
Since AB bisects DE perpendicularly, (arc AE) — (arc HD) 
and ¿EFA — ZAFD, or AF bisects the angle EFD. 
Since the angle AFB is right, FB bisects LllFG, the supple 
ment of Z EFD. 
Therefore (Eucl. VI. 3) GB : BH — GF: FH = GA : AH, 
and, alternately and inversely, AH: HB = AG : GB. 
Prop. 157 is remarkable in that (without any mention of 
a conic) it is practically identical with Apollonius’s Conics 
III. 45 about the foci of a central conic. Pappus’s theorem 
is as follows. Let AB be the diameter of a semicircle, and 
from A, B let two straight lines AE, BD be drawn at right 
angles to AB. Let any straight line DE meet the two perpen 
diculars in D, E and the semicircle in F. Further, let FG be 
drawn at right angles to DE, meeting AB produced in G. 
It is to be proved that 
AG. GB = AE. BD 
Since F, D, G, B are concyclic, Z BDG — Z BFG.