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 .