A history of Greek mathematics - Wilbourhall.org

152 APOLLONIUS OF PERGA
Adding the quadrilateral CF'H'T, we have
AH'F'Q = H'TU'R',
and similarly
By subtraction,
AHFQ = HTUR.
F'H'HF= H'TU'R' -HTUR.
Adding H'IRH to each side, we have
F'IRF == IUU'R'.
If each of these quadrilaterals is subtracted from //,
FJR'F' = JU'UR.
The corresponding results are proved in III. 5, 11, 12, 14
for the case where the ordinates through RR' are drawn to
a secondary diameter, and in III. 15 for the case where P, Q
are on the original hyperbola and R, R' on the conjugate
hyperbola.
The importance of these propositions lies in the fact that
they are immediately used to prove the well-known theorems
about the rectangles contained by the segments of intersecting
chords and the harmonic properties of the pole and polar.
The former question is dealt with in III. 16-23, which give
a great variety of particular cases. We will give the proof
of one case, to the effect that, if OP, OQ be two tangents
to any conic and Rr, R'r' be any two chords parallel to
them respectively and intersecting in J, an internal or external
point,
then RJ . Jr :
R
f J .
Jr' = OP 2 : OQ
2 = (const.).
We have
RJ. Jr = RW ^JW 2 2 , and RW 2 : JW = ARUW 2 :
therefore
AJU'W;
RJ.Jr: RW 2 = (RW 2 - JW 2 ) : RW
2 = JU'UR :
ARUW.
But RW 2 : OP
2 = ARUW: AOPT;
therefore, ex aequali, RJ.Jr: OP 2 = JU'UR :
A OPT.