422 
PAPPUS OF ALEXANDRIA 
- Props. 149, 151. 
If AB . BC =■ BD 2 , 
then (AD±DG)BD = AD. DC, 
(AD±DC) BC= DC 2 , 
A C B 0 
A D § 
and (AD ± DC) BA = AD 2 . 
Props. 152, 153. 
If AB: BG = AD 2 : DC 2 , then AB . BC = BD 2 . 
A C B D 
1 1 
A D C B 
Prop. 160. 
If AB : BG=AD : DC, then, if Abe the middle point of AC, 
BE. ED — EG 2 , 
BD.DE= AD. DC, 
EB.BD = AB.BC. 
A E D C B 
i—f i 
The Lemmas about the circle include the harmonic proper 
ties of the pole and polar, whether the pole is external to the 
circle (Prop. 154) or internal (Prop. 161). Prop. 155 is a 
problem, Given a segment of a circle on AB as base, to inflect 
straight lines AC, BC to the segment in a given ratio to one 
another. 
Prop. 156 is one which Pappus has already used earlier 
in the Collection. It proves that the straight lines drawn 
from the extremities of a chord (DE) to any point (F) of the 
circumference divide harmonically the diameter (AB) perpen 
dicular to the chord. Or, if ED, FK be parallel chords, and 
EF, DK meet in G, and EK, DF in H, then 
AH-.HB = AG:GB.