It's math. Can you explain where Leslie can put the sprinkler so each flower gets the same amount of water?

2 flowers were planted. Leslie has a sprinkler that sprays in a circle. To get the same amt, the sprinkler needs to be the same distance 4m each flower. She can put it anywhere on the line in between [view picture]. Now there's 3 flowers. What arrangements work and dont? Anything works when you circumscribe it, right? The problems continue for 4, 5, 6+ flowers. The goal is to find one generalization that will work for all of them. EXPLAIN AS FULLY AS POSSIBLE IN MATHEMATICAL LANGUAGE [w/o reference to flowers or sprinklers] HOW YOU CAN SOLVE EACH PROBLEM WITH ONE FORMULA OR GENERALIZATION. ;] Go here to see the picture. http://i14.photobucket.com/albums/a328/S...

It's math. Can you explain where Leslie can put the sprinkler so each flower gets the same amount of water?
well, in real life, if the flowers are within the circumference, its all good
Reply:So what youre asking is:





On a Euclidian surface, two points are positioned. If a third point was added, what possible places can it be situated to maintain equidistance of the original two?





Now there are three points on the Euclidian surface. Is it still possible to find the centroid that is equidistant from all three original points?
Reply:Neat question. However, it seems to imply there is always a solution for random locations. Considering the solution for three random flower locations is a single point, we can obviously have a fourth flower which does not lie on the circle, and so there would not be a solution for these four flowers.





Pete R and Goodgirl sound right to me.
Reply:If points are flowers and the radial midpoint is the sprinkler.





The points must be arranged in a way that they can be connected to form a regular polygon. In order for the radial midpoint to be equidistant from all the points, you connect the points into a regular polygon shape and circumscribe a circle that contains all of the points in the polygon. The radial midpoint will always equally distribute if it is placed directly on the midpoint of the circumscribed circle. The points must always form a regular polygon and the circumscribed circle must always contain all the points, but if these conditions are met, it will not matter how many points there are.
Reply:Of every two points along the "circle" find the line that is equidistant to the two points... just like in the first example picture. Now if all these equidistant composed lines meet at one point than your in luck -put the sprinkler there.
Reply:To find the center of any set of points you must find the centroid of all of the points. Since each problem is made of points, connect the dots to form a polygon. If you notice the pictures in the link, then all of the centers are created by drawing a line from each vertex through the midpoint of the opposite side of the polygon. (In a triangle, this would be called the median.) Generally speaking, when all points have the "median" drawn from each vertex to the opposite side, the intersection of these lines is called the centroid. This point (the centroid) is equidistant from all points in the polygon. (and therefore "waters" all of the "flowers" equally).





I hope this helps....





(addition) I thought about your comment of circumscribing points and i *THINK* this could be an answer. Maybe think about circumscribing different sets of points then find the centroid formed from the *centers* of the circles?? Just a thought...





Otherwise it is time to buy additional sprinklers.... Let me know what you think...