171
THE CONICS, BOOK YII
But A'Q 2 :A'M.MH' = AA': AH' (VII. 2, 3)
= AA':A'H
= A'M.A A': A'M. A'H,
so that, alternately,
A'M.AA':A'Q 2 = A'M. A'H: A'M. MH'
= A'H: MB'.)
Next, PP' 2 : DP' 2 = MH': MH, as above, {(3)
= MH' 2 :MH.MH',
whence PP': DD' = MH': V{ MH. MH'), (y)
and PP' 2 :{PP' + DD') 2 = MH' 2 : {MH'+ V{MH.MH')} 2 ;
(1) above follows from this relation and (a) ex aequali;
(2) follows from (a) and (y) ex aequali, and (3) from (a)
and (/3).
We now obtain immediately the important proposition that
PP' 2 + DD' 2 is constant, whatever be the position of P on an
ellipse or hyperbola (the uppey sign referring to the ellipse),
and is equal to AA' 2 + BB' 2 (YII. 12, 13, 29, 30).
For AA' 2 : BB' 2 = AA':p = A'H: AH = A'H: A'H',
by construction;
therefore AA' 2 : A A' 2 + BB' 2 = A'H: HH';
also, from (a) above,
AA' 2 :PP' 2 = A'H: MB']
and, by means of (/3),
PP' 2 : {PP' 2 + DD' 2 ) = MH': MH' + MH
= MH':HH'.
Ex aequali, from the last two relations, we have
A A' 2 : {PP' 2 + DD' 2 ) = A'H: HH'
= A A' 2 : A A' 2 + BB' 2 , from above,
PP' 2 + DD' 2 = A A' 2 + BB' 2 .
whence