ABC is a triangle with area equal to 20 . The incircle of triangle ABC has radius equal to 2 and the circumcircle of triangle ABC has radius equal to 6 . If sinA+sinB+sinC=a/b , where a and b are coprime positive integers, what is the value of a+b ?

To find the value of a + b, we first need to find the values of sinA, sinB, and sinC.

Let's start by considering the incircle of triangle ABC. The inradius of a triangle is the radius of its incircle, which is given as 2 in this case. The area of a triangle can be expressed as: Area = (Semi-perimeter)(Inradius), where Semi-perimeter = (a + b + c)/2.

We are given that the area of triangle ABC is 20. Therefore, we can write the equation: 20 = (a + b + c)/2 * 2. Simplifying this equation, we get: 40 = a + b + c.

Now let's consider the circumcircle of triangle ABC. The circumradius of a triangle is the radius of its circumcircle, which is given as 6 in this case. The circumradius can be related to the sides of the triangle using the formula: Circumradius = abc / (4*Area).

Using the given values, we can write the equation: 6 = abc / (4*20). Simplifying this equation, we get: 6 = abc / 80. Cross-multiplying and simplifying, we have: abc = 480.

We can use these equations to find the values of a, b, and c. However, since we are interested in the values of sinA, sinB, and sinC, we need to further analyze the relationship between the angles of the triangle and their corresponding sines.

Let's consider the Law of Sines, which states that for any triangle ABC with sides of lengths a, b, and c opposite to angles A, B, and C respectively, we have: sinA/a = sinB/b = sinC/c.

From this relationship, we can set up the equations: sinA/a = sinB/b = sinC/c = k, where k is a constant.

Since we found earlier that a + b + c = 40, we can write the equations:

sinA/k = a/40,
sinB/k = b/40,
sinC/k = c/40.

Multiplying these equations by k, we get:

sinA = ka/40,
sinB = kb/40,
sinC = kc/40.

We want to find the value of sinA + sinB + sinC, which is equal to:

(sinA + sinB + sinC) = (ka/40) + (kb/40) + (kc/40) = k(a + b + c)/40.

Since we know that a + b + c = 40, we have:

(sinA + sinB + sinC) = k(40)/40 = k.

Therefore, sinA + sinB + sinC = k.

To find the value of a + b, we need to find the value of k. To do that, we will use the equation abc = 480, which we derived earlier.

From sinA/a = sinB/b = sinC/c = k, we can rewrite the equation as:

(abc)/(40*40*40) = k.

Substituting abc = 480 and simplifying, we have:

480/(40*40*40) = k.

Evaluating this expression, we get:

k = 3/800.

Therefore, sinA + sinB + sinC = k = 3/800.

The value of a + b is the sum of the coprime positive integers in the fraction 3/800, which is 3 + 800 = 803.

Therefore, the value of a + b is 803.