Take some point V and draw two rays from it. Choose some other point W in between those two rays. Then, construct a line that touches both rays and passes through W.

Now, this line forms a closed triangle together with the two rays. The point W divides this line into two segments (x1, x2). What is the ratio of these two segments such that the area of the enclosed triangle is minimal?

Does this minimal area even exist?

In the picture x2 is longer than x1.  If the line is rotated a little bit counterclockwise it is easy to see that the area lost above W is more than the area gained below W.  So the picture does not show a minimized area.

If x2 and x1 are equal then a rotation in either direction will cause the amount gained to exceed the amount lost. 

I am not going to formalize this but it points to the solution:  the ratio is 1.

Posted by Jer on 2010-03-29 14:07:24