Explanation:

To understand the number of circles that can be drawn through three non-collinear points, let's consider the given points as A, B, and C.

Definition:
A circle can be uniquely defined by three non-collinear points.

Proof:
To prove that only one circle can be drawn through three non-collinear points, we need to show that the center and radius of the circle can be uniquely determined.

Construction:
Let's construct a circle by taking two arbitrary points A and B as the endpoints of the diameter. The center of the circle will lie on the perpendicular bisector of AB.

Case 1:
If the third point C lies on the perpendicular bisector of AB, then the circle is uniquely determined. The center of the circle will be the midpoint of AB, and the radius will be half the distance between A and B.

Case 2:
If the third point C does not lie on the perpendicular bisector of AB, then the circle is not uniquely determined. Two possible circles can be drawn through the three non-collinear points.

Proof of Case 2:
To prove that two circles can be drawn through three non-collinear points, let's consider the following scenario. Assume that the points A, B, and C are not collinear, and the circle with center O and radius r is uniquely determined.

Construction:
Let's construct a line passing through C and perpendicular to AB. This line intersects the perpendicular bisector of AB at a point D.

Case 2.1:
If C lies on the same side of AB as D, then the circle with center O and radius r can be drawn.

Case 2.2:
If C lies on the opposite side of AB as D, then another circle with center O' and radius r can be drawn. The center O' will be the reflection of O with respect to AB.

Therefore, in this case, two circles can be drawn through three non-collinear points.

Conclusion:
In conclusion, the correct answer is option B) 1. Only one circle can be drawn through three non-collinear points.