Choose a point outside of a circle. First, consider all of the chords that could be drawn within the circle on a line through this point. Then
- the chord that passes through the center of the circle is the longest.
- among all of the other chords, the ones closer to the chord that passes through the center are longer than ones that are further from it.
Now consider all of the line segments that could be drawn between the point outside of the circle and a point on the circle without passing through the interior of the circle. Then
- the line segment that falls on a line between the point and the center of the circle is the shortest.
- among all other line segments drawn as described, the ones closer to the shortest line segment are shorter than the ones further from it.
- except for the shortest one, these line segments come in congruent pairs, on opposite sides of the shortest line segment.
