A Chord
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This is a line segment called a diameter of the circle.
A straight line joining points on a circle is called a "chord" of that circle. If the line happens to pass through the center of the circle, then it's a "diameter" of that circle. The question asked about "points" on a circle, so two points on the circumference of that circle are being considered. (No line can join more than two points of a circle.)
Segement
There are an infinity of points 4cm from a given line. These points form 2 lines parallel to and either side of the original line. Equally, there are an infinite number of points 4cm from a given point on the original line. These points lie on the circumference of a circle radius 4cm with its centre at the given point. There are only 2 points that fulfil both conditions. These points are found on the circumference of the circle where a diameter perpendicular to the original line and passing through the given point meets the circumference of the circle. These two points are also where the two parallel lines form tangents with the circle.
Step I: Show that both points are outside the smaller circles. Possibly by showing that distance from each point to the centre of the circle is greater than its radius. Step 2: Show that the line between the two points touches the circle at exactly one point. This would be by simultaneous solution of the equations of the line and the circle.