A history of Greek mathematics - Wilbourhall.org

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418 PAPPUS OF ALEXANDRIA
This is equivalent to the general relation between four
points on a straight line discovered by Simson and therefore
wrongly known as Stewart's theorem
AD 2 . BC+BD
2 . CA
+ CD 2 . AB + BC.CA .
AB
= 0.
(Simson discovered this theorem for the more general case
where D is a point outside the line ABC)
An algebraical equivalent is the identity
(d _ a f (b-c) + (d - b) 2
(c-a) + (d - c ) 2 (a - b)
+ (b — c) (c — a) (a — b) = 0.
Pappus's proof of the last-mentioned lemma is perhaps
worth giving.
A c D B
C, D being two points on the straight line AB, take the
point F on it such that
Then FB :
FD:DB = AC:CB. (1)
BD = AB :
BC,
and (AB-FB) : (BC-BD) = AB.BC,
or
AF:CD = AB:BC,
and therefore
AF.CD:CD 2 = AB:BC. [2)
From (1) we derive
AG
and from (2)
. DB
or
We have now to prove that
2 = FD. DB,
^. CD 2 = AF.CD.
AD 2 + BD.DF= AC 2 + AC.CB + AF.CD,
AD 2 + BD.DF= CA.AB + AF.CD,