Brachistochrone Problem

This problem, one of the most famous in the history of mathematics, is known as the brachistochrone problem. It asks: Given two points at different heights on a plane, what is the shape of the wire down which a bead will slide (without friction), under the effect of gravity, from the upper point to the lower point in the shortest possible time? In 1638, Galileo explored this problem in his work Two New Sciences. His initial approach involved identifying a straight line from point A (the upper point) to point B (the lower point on a vertical line) as the path for the bead to travel. He calculated the time required for the bead to travel along this straight line. However, Galileo later showed that the bead would actually reach point B more quickly if it followed a path composed of two segments, AC and CB, where point C lies on an arc of a circle.