TITLE: What is the best strategy to find 2 circles whose total area is maximum in a triangle?
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: 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; http://www.cut-the-knot.org/Curriculum/Geometry/Malfatti.shtml 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.