Given three concentric (having the same center) circles. Their radii are 1,2 and 3.
Place three points A,B,C - each on a different circle to get a triangle ABC with a maximal area.

What is this area?

without loss of generality we can place the point on the circle of radius 1 at (0,1), thus the remaining points can be considered at
(2cos(x),2sin(x)) and (3cos(y),3sin(y))
with 0<=x,y<2Pi
the area of this triangle is given by
6sin(x-y)+3cos(y)-2cos(x)
Using Mathematica's Maximize function under the contstraints
0<=x,y<2Pi
it gives the maximum area of 9.8094 with x=3.95901 and y=5.77539

bellow is a link to a picture of the 3 circles along with this triangle

http://i57.photobucket.com/albums/g203/bdiddycombes/Perplexus%20Images/BigTriangle.jpg

Posted by Daniel on 2010-09-09 17:34:17