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