How Many Circles Can Pass Through 3 Noncollinear Points?

In geometry, it is a well-established principle that exactly one circle can pass through any three noncollinear points. This is based on the concept of a circumscribed circle in triangle geometry.

In this article, we will explore why exactly one circle can pass through three noncollinear points and how the circle is constructed.

What Does It Mean for a Circle to Pass Through 3 Points?

For a circle to pass through three points, each of those points must lie on the circumference of the circle. The circle is said to be circumscribed around the triangle formed by these three points. A circle can be circumscribed around any triangle, and the center of this circle is known as the circumcenter.

The three points must be noncollinear, meaning they do not lie on a straight line. If the points are collinear (i.e., lie on the same straight line), it is impossible to form a circle that passes through all three points.

Why Can Only One Circle Pass Through 3 Noncollinear Points?

Given three noncollinear points, you can always find one unique circle that passes through them. Here’s why:

1. The Circumcenter:
   • The center of the circle that passes through three noncollinear points is known as the circumcenter. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle formed by the three points intersect.
   • Since the perpendicular bisectors of any triangle’s sides intersect at only one point, the circumcenter is unique, and thus the circle that passes through the three points is also unique.
2. Equidistant Property:
   • The circumcenter is equidistant from all three vertices of the triangle. This means that the distance from the circumcenter to each of the three points is constant, which defines the radius of the circumscribed circle.
   • The existence of this unique center and radius means that there is only one circle that can pass through all three points.

Construction of the Circumscribed Circle

To construct a circle that passes through three noncollinear points, follow these steps:

1. Find the Perpendicular Bisectors:
   • Draw the perpendicular bisector of one side of the triangle formed by the three points.
   • Repeat for the other two sides of the triangle.
2. Locate the Circumcenter:
   • The point where all three perpendicular bisectors intersect is the circumcenter. This point is equidistant from all three vertices of the triangle.
3. Draw the Circle:
   • Use the circumcenter as the center and the distance from the circumcenter to any of the three points as the radius. Draw the circle with this center and radius, and it will pass through all three points.

Conclusion

In conclusion, exactly one circle can pass through any three noncollinear points. This circle is known as the circumscribed circle of the triangle formed by those points, and its center is the circumcenter, which is equidistant from all three points. The unique properties of the perpendicular bisectors and the circumcenter ensure that there is only one possible circle that can pass through these three points.

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