For four points A, B, C, D lying on a common line in this order, we can consider the set of points X such that the angles AXB and CXD are equal. In general, this set forms a circle, known as the Apollonian Circle. One may ask how this set changes if one of the points A, B, C, D does not lie on the line containing the other three points. It turns out that instead of a circle, we obtain a curve described by a cubic equation. In my paper I study geometric and algebraic properties of such curves. I also show how to construct them and describe using barycentric coordinates. Using the derived theory, I also prove numerous complex theorems. In doing so, I show that cubic curves, like lower-degree curves, serve as a highly useful tool in solving geometric problems.