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.

**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: \( \textbf{A. There is only one unique circle that can be circumscribed about a triangle.} \)