Consider the competing straights:

r: a1x + b1y + c1 = 0
s: a2x + b2y + c2 = 0,

They intersect at one point Q.

If P(x, y) is any point in any of the bisectors, PQ then P equidist of r and s:

Considering the positive sign, we get a bisector; considering the negative sign, we get the other one. Let's look at an example:

If r: 3x + 2y - 7 = 0 and s: 2x - 3y + 1 = 0, so your bisectors are:

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