406 
PAPPUS OF ALEXANDRIA 
Props. 32, 34 are alternative proofs based on other lemmas 
(Props. 31, 33 respectively). The algebraic equivalent may be 
stated thus : if ax — by, then - = ^ ^. 
V {x + y){a-y\ 
III. Props. 35, 36. 
If A B . BE = CB . BD, then AB -. BE = DA . AG-.CE.ED, 
and CB : BD = AC . CE: AD . DE, results equivalent to the 
following; if ax — by, then 
a _ (a — y ) (a — b) b _ (a — b)(b — x) 
x ~ (b-x) (y-x) lint y ~~ (a-y) (y-x) m 
IV. Props. 23, 24, 31, 57, 58. 
A B C E D 
1 1 1 1 
If AB = CD, and E is any point in CD, 
AO . CD = AE. ED + BE. EC, 
and similar formulae hold for other positions of E. If E is 
between B and G, AG.GD — AE.ED — BE.EC\ and if E 
is on AD produced, BE. EC = AE.ED + BD . DC. 
V. A small group of propositions relate to a triangle ABC 
with two straight lines AD, AE drawn from the vertex A to 
points on the base BC in accordance with one or other of the 
conditions (a) that the angles BAG, DAE are supplementary, 
(b) that the angles BAE, DAG are both right angles or, as we 
may add from Book VI, Prop. 12, (c) that the angles BAD, 
EAG are equal. The theorems are: 
In case (a) BC .CD-.BE .ED = GA 2 : AE 2 , 
„ (b) BG. CE: BD . DE = GA 2 : A D 2 , 
„ (c) DC. CE: EB . BD — AG 2 : AB 2 .