A history of Greek mathematics - Wilbourhall.org

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To prove (1)
R'l 2 : IW-H'Q 2 : QH
THE CONICS, BOOK III 155
we have
= 2 AH'F'Q :
AHFQ
= H'TU'R' :
Also R'T 2 : TR = 2 R'U' 2 : UR = AR'U'W 2 :
and jRT a : Ti? 2 = TW 2 : TW = ATH'W 2 :
HTUR
(III. 2, 3, &c).
A220TT,
A TWIT,
so that R'T 2 :TR 2 = ATH'W' - AE'CHF: ATi^TF- A-RETTF
= H'TU'R':HTUR
= R'l 2 : i7? 2 , from above.
To prove (2) we have
RV 2 : 7iT 2 = RU 2 : R'U' 2 = ARUW: AR'U'W,
and also
= HQ 2 : QH' = AHFQ 2 :
so that
RV 2 : VR'
AH'F'Q
= HTUR * :
H'TU'R',
2 = HTUR + ARUWiH'TU'R' + AR'U'W
= ATHWiATHW
= TF 2 : TIP
= RO 2 : OR'
2
2 .
Props. III. 30-6 deal separately with the particular cases
in which (a) the transversal is parallel to an asymptote of the
hyperbola or (6) the chord of contact is parallel to an asymptote,
i.e. where one of the tangents is an asymptote, which is
the tangent at infinity.
Next we have propositions about intercepts made by two
tangents on a third : If the tangents at three points of a
parabola form a triangle, all three tangents will be cut by the
points of contact in the same proportion (III. 41) ;
if the tangents
at the extremities of a diameter PP' of a central conic
are cut in r, r' by any other tangent, Pr . P'r' = CD 2 (III. 42)
if
the tangents at P, Q to a hyperbola meet the asymptotes in
* Where a quadrilateral, as HTUR in the figure, is a cross-quadrilateral,
the area is of course the difference between the two triangles
which it forms, as HTW ^ RUW.