The Surfer and the Spotter
Alex wants to be closest to the points of the island so towards the vertices of the triangle. I used the regular polygon to construct an equilateral triangle as the island. Then I constructed point A in the triangle and this was Alex. Next I constructed segments from point A to each of the corners of the triangle C,D,E. Then use the algebra section of GeoGebra I added together the lengths of AC, AD, and AE. We want Alex's path to all three corners to be minimized. So I began to move A around unit I found the minimum distance. I noticed as I moved along the outline of the triangle that the distance was the same and as I moved more towards the center of the triangle the distance began to decrease. The closer I got to the center of the triangle the smaller the distance got. I finally found the smallest distance. Once I found this distance I observed that all three segments from the points of the triangle to the center were equal. This point is the center of the triangle. It is also the incenter, the circumcenter, the orthocenter, and the centroid. The incenter is the point of intersection of the angle bisectors. The circumcenter is the point of intersection of the perpendicular bisectors. The orthocenter is the point of intersection of the altitudes, and the centroid is the point of intersection of the medians. Therefore from this we can conclude that the Alex should be at the center of the island so that he is the shortest distance away from the corner of the islands which is the same to each corner.
Beth wants to be the closest to the beaches of the islands. On the sketch I plotted point B for Beth. I constructed three perpendicular lines through point B to the sides of the triangles. I made these line perpendicular lines because I had the previous knowledge that the shortest path from a point to a line is perpendicular to the line. Then I constructed three segments from the intersections of the sides of the triangles and the perpendicular lines. Next I found the sum of the lengths of these distances. I did the same as with Alex and began to move Beth around the island to find the minimum distance to the beaches. When I moved the point around the distance was always the same. At first I thought that something was wrong with my sketch but that was not the case. Whether she was in the middle of the island, in the corner, or on the beach it is the same distance. As Beth approaches any corner of the island, the three paths converge into a single path that is an altitude in the triangle. An altitude is a line segment through a vertex and perpendicular to the opposite side of the triangle. The sum of the lengths of the three paths is equal to the lengths of an altitude, regardless of where Beth is located.