1) f(x) = (A*x)/(C*x + D).

2) g(x) = (A1*x

Then the line L

Construct this conic.

The construction illustrated in the above figure is carried out as follows.

1] To show that the envelope is a conic reduce it to a Chasles-Steiner envelope for lines joining points in homographic relation.

(See Chasles_Steiner_Envelope.html ). For this show that there is a third vector

2] This needs some calculation but is not problematic to show.

3] It is also relatively easy to show that the vector

4] According to Chasles-Steiner the enveloping conic is tangent to the lines generated by {

5] Further it is easy to find the point of contact f

6] Equaly easy is determined the point g

The formulas above give hints on how this is done.

7] It is also easy to find the tangents parallel to the second line, intercepting the x-axis at f

8] Then the parallelograms circumscribed and inscribed in the conic and starting at f

9] The conic is determined as the member of the bitangent family determined by the triangle f

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