(b) If the triangle has maximum area, it is isosceles. Let the base chord be a distance from the center of the circle. Then the altitude of the triangle is and the base is . The area is then , which has a critical point ar . It follows that each side of the triangle is , so it is equilateral. The area is .
(b) The pipeline would run from to to .
(c) Let be the reflection of in the river; then the shortest pipeline would have the same length as the straight line from to , which meets the river at point between and . We can now use similar triangles to find , which is the same as in (a).