THE COLLECTION. BOOK VII 
417 
The other problem (Prop. 117, pp. 848-50) is, as we have 
seen, equivalent to the following: Given a circle and three 
points D, E, F in a straight line external to it, to inscribe in 
the circle a triangle ABC such that its sides pass severally 
through the three points D, E, F. For the solution, see 
pp. 182-4, above. 
(e) The Lemmas to the Plane Loci of Apollonius (Props. 
119-26, pp. 852-64) are mostly propositions in geometrical 
algebra worked out by the methods of Eucl., Books II and VI. 
We may mention the following : 
Prop. 122 is the well-known proposition that, if D be the 
middle point of the side BG in a triangle ABC, 
BA 2 + AC 2 = 2 (AD 2 + DC 2 ). 
Props. 123 and 124 are two cases of the same proposition, 
the enunciation being marked by an expression which is also 
found in Euclid’s Data. Let AB: BG be a given ratio, and 
A dec b 
A D c B E 
let the rectangle CA .AD be given; then, if BE is a mean 
proportional between DB, BG, ‘the square on AE is greater 
by the rectangle GA . AD than in the ratio of AB to BG to the 
square on EG\ by which is meant that 
A Ti 
AE 2 = GA. AD+ ~ .EC 2 , 
or (AE 2 - GA . AD): EG 2 = AB : BG. 
The algebraical equivalent may be expressed thus (if AB=a, 
BG = h, AD = c, BE = x): 
T£ // TT in (a + x) 2 -(a-b)c a 
It x — v(a~c)b, then — ; ._ 7 -o = 
v ' (x + b) 2 b 
Prop. 125 is remarkable: If C, D be two points on a straight 
line AB,