A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE CONIGS, BOOK I 143
(2) in the hyperbola or ellipse, ARUW = the difference
between the triangles GFW and CPE.
(1) In the parabola ARUW: AQTV = RW 2 : QV
But, since
therefore
TV = 2PV, AQTV=CJEV:
A RUW = O EW.
2
= PW:PV
= EJEW:njEV.
(2) The proof of the proposition with reference to the
central conic depends on a Lemma, proved in I. 4 1
, to the effect
that, if PX, VY be similar parallelograms on CP, GV as bases,
and if VZ be an equiangular parallelogram on Q V as base and
such that, if the ratio of CP to the other side of PX is m, the
ratio of QV to the other side of VZ is m .
p / ' PP', then VZ is
equal to the difference between VY and PX. The proof of the
Lemma by Apollonius is difficult, but the truth of it can be
easily seen thus.
By the property of the curve, QV 2 : CV ^CP = 2 2 p: PP'
;
PP'
therefore CV 2 - CP 2 = ~. QV 2 .
P
Now LJPX = p. CP 2 /m, where /i is a constant depending
on the angle of the parallelogram.
Similarly
CJVY=a.CV 2 /m, and E3VZ = a
It follows that
DFF- DPI = o VZ.
——QV 2 /m.
r
p
Taking now the triangles GFW, CPE and R UW in the
ellipse or hyperbola, we see that GFW, CPE are similar, and
RUW has one angle (at W) equal or supplementary to the
angles at P and V in the other two triangles, while we have
whence QV: VT = (p
: PP')
and, by parallels,
QV*:CV.YT =