Extreme Yin-Yang 2

At first this problem seemed to be a particularly difficult maximization problem, where the 3 circles "can be of any radii" that would fit inside a circle of radius $9$ . But if it's specified that two of the circles are tangent at the center of the large circle, then it's a straightforward computation of the red area. The radius of the large circles being $4.5$ , it doesn't take long to determine that the radius of the small circle is $3$ . From this, the area of the red area is

$\dfrac { 1 }{ 2 } (6)(9)-\dfrac { 1 }{ 2 } { 3 }^{ 2 }(2ArcTan(\dfrac { 4.5 }{ 6 } ))-2(\dfrac { 1 }{ 2 } { 4.5 }^{ 2 }ArcTan(\dfrac { 6 }{ 4.5 } ))=2.43076...$

This is not a complete solution. For the complete solution, subscribe to this discussion and in 4 days, I will post the full solution to solving this problem without the degree measure on the left if no one else has posted it (this way others have a chance to post their brilliant solutions to solving this problem the hard way).

We start by drawing a triangle connecting the three circle centers. Using law of cosines, we find the dimensions to be 15/2, 9, and 15/2. Then using $1/2absin(c)$ , the area of the triangle is 27. Next, we have to subtract the areas of the circles inside the triangle by taking their area and multiplying it by its arc measure/360. After some messy calculations, we find the area of the red sector to be $\boxed{2.43}$ .