Problem 164
The skipper of a boat knows that the water is too shallow to bring his boat inside a “danger semicircle” that goes from lighthouse to lighthouse. How can the angle between his sightings of the two lighthouses help him decide when he is entering the danger zone?
Solution
The skipper can apply the Thales’ theorem, which says that if A, B, and C are points on a circle, and the line AC is a diameter of the circle, then the angle ABC is a right angle.
The distance between the two lighthouses is a diameter, and they correspond to points A and C. When the skipper arrives to any point B on the semicircle of danger, the angle formed by his sightings of both lighthouses will be 90°.
At that moment he’ll know that he’s entering the semicircle.