There is one and only one circle passing through three givennon-collinear points.

This statement is true. Here's why:

Step 1: Understand the terms

• Non-collinear points: These are points that do not lie on the same line. In other words, you cannot draw a straight line that passes through all three points.
• Circle: A round shape consisting of all points in a plane that are a certain distance from a certain point, the center.

Step 2: Visualize the problem Imagine three points in space that are not on the same line. You can intuitively see that there is a unique way to draw a circle that passes through all three points.

Step 3: Prove the statement To prove this mathematically, you can use the fact that a circle is defined by its center and radius. The center of the circle passing through three non-collinear points is the intersection of the perpendicular bisectors of the segments joining the points. Since the points are non-collinear, these bisectors are distinct and intersect at a unique point, which is the center of the circle. The radius is the distance from this center to any of the three points. Therefore, there is only one circle that can pass through three given non-collinear points.