What is the least possible radius of a circle through two points and ?

Question

Homi · Accepted Answer

Understand the Question

Any circle passing through two points $A$ and $B$ must have its center equidistant from both points ( $CA = CB = r$ ).
The locus of all points equidistant from $A$ and $B$ is the perpendicular bisector of the segment $AB$ .
The distance from the center to either point is minimized when the center is chosen as the midpoint of $AB$ , making segment $AB$ the diameter of the circle.

Step 1 · Position of the Circle's Center

Let $C$ be the center of a circle passing through points $A$ and $B$ .

$CA = CB = r$

Since $C$ is equidistant from $A$ and $B$ , $C$ must lie on the perpendicular bisector of the segment $AB$ . Diagram 1

Step 2 · Find the Minimum Radius

The radius $r = CA$ is minimized when the center $C$ is at the closest possible point to $A$ on the perpendicular bisector.

This occurs when $C$ is the midpoint of the segment $AB$ , making $AB$ the diameter of the circle.

$\text{Least Radius} = \dfrac{\text{Length of } AB}{2} = \dfrac{AB}{2}$

Answer

$\dfrac{AB}{2}$

Common Mistakes

Confusing Radius with Diameter: Stating the least radius is $AB$ rather than $\dfrac{AB}{2}$ , mistakenly treating the segment length as the radius instead of the diameter.
Unique Circle Misconception: Assuming only one circle can pass through two points; infinitely many circles can pass through $A$ and $B$ , but the circle with diameter $AB$ has the smallest possible radius.

Question 4