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,