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I am looking for the best strategy to find 2 circles whose total area are maximum in a triangle. I tried it on an example as seen Figures below.

Strategy $1$:

If I draw a big circle that touchs to 3 sides and then to draw 3 circles next to the big circle. Then select the biggest one in three of them. As seen in Figure 1 that I found Max Area of 2 circles: 14.18+4.56=18.74

Strategy $2$:

To drew 2 circles that each one touched to 2 sides of triangles and also touching each other as seen in Figure-2. I tried to extend total area that what I got in Strategy 1 but I could not.

Of course I know my example is not proof that Strategy 1 is general solution of the problem. I just tried to show what I did till now to solve the optimum problem.

Could you please help me to find the best strategy and to proof it for that optimum problem? Note:If there is a general strategy proof for total max area for n circles in a triangle, It can be wonderful. Thanks for your answers and your time.

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Problems like this are hard. A look at will show the range of configurations involved. The way I would try to show that Strategy 1 is optimal for your particular problem is to take a general version of Strategy 2 and show that things always get better if you enlarge the larger circle, shrinking the smaller one as required. No guarantees that this will work. – Ross Millikan Apr 12 '12 at 15:30
up vote 2 down vote accepted

The problem with three circles has a long and interesting history. It goes by the name, Malfatti's problem, and you will find much by typing said keywords into the internet; is one nice discussion.

I suspect that if you hunt around in the Malfatti literature you will find either some methods that can be applied to the 2-circle problem or else that someone has already done it for you.

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