Mathematics has always been concerned with extending existing concepts to more comprehensive cases, I am a big fan of what I would call "imaginary geometry" might be a bad name, I don't know its basic name, let me explain a bit...
The concept of radical axis emerged as a generalization of the concept of a line that connects two intersection points of two circles, which can be created for any two circles that are not concentric in the plane.
The generalization was not in vain, as the alternative line carried many radical axis properties, for example, the power its points is equal to the two circles and The convergence of the three lines of this type at one point and other characteristics.
I believe that this idea exists over and over again in geometry, and that what we called relative cases that cannot achieve the basic definition can be extended to include those cases using deeper and more general concepts.
In the process of enumerating and explaining what I came to in this question, it will be very long, but here is one example to understand what I mean:
It is known that it is impossible to create a parabola that touches two circles that are spaced inside and each of them touches it at two points, but I found that it is possible to create a parabola that carries most of its geometric and algebraic properties, decisively that can be created depending on three properties enjoyed by the parabola that touches two circles, each of which touches it In two points:
The distance between the midpoint of the distance between the centers of the two circles and the radical axis of these two circles is equal to the distance between the focus and the guide of the parabola.
The focus of the parabola is midway between the centers of similarity of the two circles.
The axis of symmetry of the parabola is the same as the axis of symmetry of the two circles.
As for the properties 1 and 2, I came to them myself and it would be a pity if they were discovered in advance because I consider them among the discoveries that are distinguished for me.
For example it keeps the property I recently posted in my question:
The sum of the vector lengths of the tangents of two circles in a conic section look at the picture:
In any case, what I am looking for in my question is to know whether this type of thing has already been studied and there are many concepts that have been generalized in a similar way, or is it still a topic at the beginning of the road, are there researches or books that talk about this subject in depth?
Here is a list of some concepts that I suspect can be generalized similarly:
• Generalizing the problem of Apollonius to the impossible relative cases.
• Draw the intersection line(s) that connects the intersection points of two divergent conic segments.
• Creating a circle that passes through four points that do not form the vertices of a circular quadrilateral
• Imaginary Euclidean triangles do not satisfy the triangle inequality.
• Newton-Gauss line that connects the midpoints of the diagonals of a complete quadrilateral when the vertices of the original quadrilateral are any points in space.
Please mention suggestions for generalizable concepts in the comments, and mention possible books or references on this subject
Also, was what I reached about the parabola tangent to two circles, each of which touches it at two points, previously discovered, and was the generalized parabola that I mentioned previously discovered? Thank you.
This is a link that includes my two theorems in Arabic: https://drive.google.com/file/d/1iegrVBA0nwm30g-x3MqiPJpk7SUJIHBQ/view?usp=drivesdk