Which of the following statements is true regarding the relationship between circles and triangles? A. There is only one unique circle that can be circumscribed about a triangle. B. There are many circles that can be inscribed in a given triangle. C. There are many circles that can be circumscribed about a triangle. D. There are many triangles that can be inscribed in a given circle.

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**Step 1. Analyze Statement A**

Given any triangle, a circle that passes through all three vertices is called the circumcircle. One of the fundamental theorems in geometry shows that the perpendicular bisectors of the sides of a triangle are concurrent. Their point of concurrency, called the circumcenter, is the center of the unique circumcircle. Thus, there is exactly one circle that can be circumscribed about a given triangle. In mathematical terms, if a triangle has vertices $$A$$, $$B$$, and $$C$$, then the perpendicular bisectors of segments $$\overline{AB}$$, $$\overline{AC}$$, and $$\overline{BC}$$ intersect at one unique point $$O$$. The circle with center $$O$$ and radius $$OA$$ (or equivalently $$OB$$ or $$OC$$) is the unique circumcircle.  
 
**Step 2. Analyze the Other Options**

- **Statement B:** “There are many circles that can be inscribed in a given triangle.”  
  In any triangle, the circle tangent to all three sides is called the incircle, and it has a unique location. The angle bisectors of the triangle are concurrent at the incenter, which is the center of the incircle. Thus, there is only one incircle, not many.

- **Statement C:** “There are many circles that can be circumscribed about a triangle.”  
  As described in Step 1, the circumcircle is unique. There is not more than one circle that passes through all three vertices of a triangle.

- **Statement D:** “There are many triangles that can be inscribed in a given circle.”  
  It is true that if you are given a circle, you can form infinitely many triangles by selecting any three non-collinear points on the circle. While this statement is correct, the well-known relationship between circles and triangles usually refers to the unique circumcircle associated with a given triangle.

**Step 3. Determine the Best Answer**

Both Statements A and D are true. However, the special and often emphasized geometric relationship is that every triangle has one and only one unique circumscribed circle (the circumcircle). This is a fundamental fact in Euclidean geometry.

Thus, the statement that best captures this unique relationship is:

$$ 	extbf{A. There is only one unique circle that can be circumscribed about a triangle.} $$
There is only one unique circle that can be circumscribed about a triangle.

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