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A line through a circle that does not go through the center of the circle is a secant line. A line through a circle that does go through the center is still a secant line, by the way. Compare this to a line segment that has its two endpoints on the circumference of the circle. That line segment is a cord of the circle. If that cord of the circle passes through the center of the circle, then the cord is a diameter of that circle.
Circle with a line through it is square yards, a square with a line through it is square feet
There is no name for it except "A line perpendicular to a line segment and passing through its midpoint".
A circle with a line through it identifies codes that are not used with a modifier.
The shortest path is a line perpendicular to the given line that passes through the given point.