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,