4 lines create conics
The equation of a line is of first degree. The product of two lines is of second degree and is some form of conic. This product set to zero would be a degenerate conic formed of two lines. The sum of two such terms is, in general, a non-degenerate conic. In this way we build more complicted graphs from simple ones like lines, and later conics.

Apple Computer's new Grapher program was used to make these pictures, but any program that will graph implicit equations will do.

In this picture 4 line functions L1, L2, L3, L4 create functions representing 2 conics S1 and S2 using S1 = L1 L2 + L3 L4 and S2 = L1 L2 - L3 L4 . At the intersection of lines 1 and 3, both L1 and L3 are zero, making both S1 and S2 zero, so that the conics go through that intersection.

Similarly they go through the intersections of lines 1 and 4, lines 2 and 3, lines 2 and 4 (but not 1 and 1 or 3 and 4)

These 4 lines intersect in 6 points, 4 of which are on the two conics. The general equation L1 L2 + n L3 L4 = 0 represents all conics through these four points.

By writing these lines in normal form, we can prove that for a point on either of these conics, the product of the signed distances to the lines 1 and 2 is either equal or the negative of the distances to lines 3 and 4.

These conics are circumconics to the quadrilateral whose pairs of opposite sides are lines 1,2 and 3,4.

Here the parallel lins are the same. Now the graphs are tangent to lines 4 and 3 at their intersection with line 1.

The blue conic is now S1 = (L1)^2 - L3 L4. The square in one term produces a double point at the intersections of lines 1 and 3 as well as 2 and 4.

Similarly for the red conic.