28.1.12 In the plane, given 5 points, where no three points are collinear, and no four points are concyclic. If a circle passes through three of these points, and the other two points are respectively inside and outside the circle, then it is called a good circle. Let the number of good circles be $n$, try to find the possible values of $n$.


Consider any two points $A, B$ among 5 points, and let the remaining three points be $C, D, F$.
If $C, D, E$ are on the same side of line $AB$, without loss of generality, assume $\angle ACB < 180^{\circ}$, in this case, the circle $ABD$ is the unique good circle;
(3) $\angle ACB + \angle AEB < 180^{\circ} < \angle ADB + \angle AEB$, in this case, the circles $ABC, ABD, ABE$ are all good circles;

The above discussion shows that for any two given points, the number of good circles is either 1 or 3.
5 points can form 10 pairs, and there is at least one good circle passing through each pair, so there are at least 10 good circles (counted with repetition). Since each good circle passes through exactly 3 pairs, there are at least 4 different good circles, i.e., $n \geqslant 4$.

With 5 points, 10 line segments can be formed, each of which is a chord of an odd number (1 or 3) of good circles. The sum of 10 odd numbers is even, and each good circle has exactly 3 chords, so $n$ is even.

If $n \geqslant 6$, then at least $\frac{3 \times 6 - 10}{2} = 4$ line segments are common chords of 3 good circles. 4 line segments have 8 endpoints, so there must be two line segments with a common endpoint, say $AB$ and $AC$. Thus, the circles $ABD$ and $ACD$ are both good circles, so there are at least 2 good circles passing through $A$ and $D$, and of course, there must be 3. Similarly, there are 3 good circles passing through $A$ and $E$. By symmetry, assume the shortest line segment among $AB, AC, AD, AE$ is $AB$. Then $\angle ACB, \angle ADB, \angle AEB$ are all acute angles. From the previous discussion, if $C, D, E$ are on the same side of line $AB$, then there is only one good circle passing through $A$ and $B$; if $C, D, E$ are on different sides of line $AB$, then since the sum of any two angles among $\angle ACB, \angle ADB, \angle AEB$ is less than $180^{\circ}$, there is also only one good circle passing through $A$ and $B$. Contradiction!
In conclusion, the number of good circles $n = 4$.