THE CONICS, BOOK VII 
173 
conjugate diameters in an ellipse or conjugate hyperbolas, and 
if the tangents at their extremities form the parallelogram 
LL'MM', then 
the parallelogram LL'MM' = rect. AA'. BB'. 
The proof is interesting. Let the tangents at P, D respec 
tively meet the major or transverse axis in T, T'. 
Now (by VII. 4) PT 2 : CD 2 = NT: CN; 
therefore 2 A GPT: 2 A T'PC = NT: CN. 
L 
= CP: PT', by similar triangles, 
= (CL): 2 AT'PC. 
That is, (CL) is a mean proportional between 2 A GPT and 
2 A T'PC. 
Therefore, since V{NT .CN) is a mean proportional between 
NT and CN,