A history of Greek mathematics - Wilbourhall.org

THE COLLECTION. BOOK VII 423
Since AB bisects BE perpendicularly, (arc AE) — (arc AD)
and Z.EFA = lAFD, or AF bisects the angle EFD.
1
JF
A [— *"* H/ '^ ^^^^^?
;)b ..--**
X — *" * ^^^
'k
Since the angle J.i^B is right, FB bisects AHFG, the supplement
of Z EFD.
Therefore (Eucl. VI. 3) GB :
and, alternately and inversely, AH :
BLI = GF : FH
HB = AG :
= GzL : ^Itf,
GB.
Prop. 157 is remarkable in that (without any mention of
a conic) it is practically identical with Apollonius's Conies
III. 45 about the foci of a central conic. Pappus's theorem
is as follows. Let AB be the diameter of a semicircle, and
G'x -A
from A, B let two straight lines AE, BD be drawn at right
angles to AB. Let any straight line DE meet the two perpendiculars
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.