A history of Greek mathematics - Wilbourhall.org

. EC
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 arid 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 sever ally
through the three 'points D, E F. For the solution, see
y
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 BC 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 : BC be a given ratio, and
A DEC B
let the rectangle CA .
AD
be given ; then, if BE is a mean
proportional between DB, BC, the square on AE ' is greater
by the rectangle CA .
square on EC\ by which is meant that
AE 2
or (AE 2 - CA .
AD than in the ratio of AB to BC to the
A R
= CA.AD+ ~ c
AD)
:
2 ,
EC = AB 2 :
BC.
The algebraical equivalent may be expressed thus (if AB=a,
BC = b, AD = c, BE=x):
jo /, zt ,1 (aTx) 2 -(a — b)c a
It x = v(a — c)b, then ; _: —— = -*•
v ;
(x + bf b
Prop. 125 is remarkable :
line AB,
If C, D be two points on a straight
^
Aft
/ID
AD 2 + . DB = 2 AC 2 + AC.CB+ == . CD\
1823,8
E e