Euclid's Elements - Book III, Proposition 7
Choose a point on a given diameter of a circle, but not the center. Then consider all of the line segments with this endpoint and another endpoint on the circle. Then - the longest of these line segments is the one that passes through the center of the circle. - the shortest of these line segments is the portion of the given diameter that does not include the center. - among all other line segments drawn from the point to the circle, the ones closer to the line segment that passes through the center are longer, and the ones further from the line segment through the center are shorter. - these line segments drawn from the point but not along the diameter come in congruent pairs, on opposite sides of the diameter.
Move points A, B, and P to explore the relationships between the line segments.