Final answer:
No lines or circles can be drawn equidistant from all four given points in a plane that are not all collinear and not all lying on a circle.
Step-by-step explanation:
Given four points in the plane that are not all collinear and not all lying on a circle, the number of lines and circles that can be drawn equidistant from all four points is none.
A single line can only be equidistant from, at most, two points unless they are all collinear, which is not the case here. Similarly, a single circle can only be equidistant from all points if the points lie on the circumference of that circle. Since the points do not all lie on a single circle, such a circle cannot exist.
The concept of points being equidistant from a center is a foundation for the definition of a circle. In contrast, an ellipse has two foci, and for any point on the ellipse, the sum of the distances to the two foci is constant. However, with four arbitrary points, you cannot define an ellipse or any other conic section that is equidistant to all four points unless specific conditions are met.
In conclusion, no lines or circles can satisfy the condition of being equidistant from all four given points unless additional criteria are satisfied, such as the four points form a rectangle (in which case the circumcircle would be equidistant to all points).