Final answer:
The maximum number of points at which a circle can intersect a triangle is six, given that each side of the triangle can intersect the circle twice; once when entering and once when exiting.
Step-by-step explanation:
The question asks what is the maximum number of points of intersection between a circle and a triangle. The maximum number of points at which a circle can intersect a triangle is six. This is because a triangle has three sides, and each side can intersect the circle at a maximum of two points - once when entering the circle and once when exiting the circle. Visualize or draw a circle with a triangle outside of it, and then move the triangle so that each side just touches the circle. Each side would then touch the circle at one point, resulting in three intersection points. Now, imagine increasing the size of the triangle or moving it closer to the center of the circle. Each side of the triangle would now cut across the circle, creating two points of intersection per side - one where the side enters the circle and one where the side exits the circle. So, with three sides creating two intersection points each, the maximum number of intersection points is six.