Analysis: According to Pythagorean theorem, the triangle is a right triangle, the radius of the inscribed circle of the triangle is 1, and the intersection points of the circle and the straight line at different positions range from none to at most four.

Answer:

Solution: ∫3 square +4 square =25, 5 square =25,

A triangle is a right triangle,

Let the radius of the inscribed circle be r, then

(3+4+5)r=？ ×3×4,

The solution is r= 1,

So it should be divided into five situations:

When an edge is separated from a circle, there are 0 intersections.

When an edge is tangent to a circle, there are 1 intersections.

When an edge intersects a circle, there are two intersections.

When a circle is inscribed with a triangle, there are three intersections.

When two edges intersect a circle at the same time, there are four intersections.

Therefore, the number of common points may be 0, 1, 2, 3 and 4.

So choose C.

Comments: This question examines the intersection of line segments and circles and needs to consider all possible situations. Finding the radius of inscribed circle first is the key to solve the problem.