Equilateral triangle ABC has a circumcircle Γ with center O and circumradius 10. Another circle Γ1 is drawn inside Γ such that it is tangential to radii OC and OB and circle Γ. The radius of Γ1 can be expressed in the form ab√−c, where a,b and c are positive integers, and b is not divisible by the square of any prime. What is the value of a+b+c?

To find the radius of circle Γ1, we can use the relationship between the radii of the circumcircle and the incircle of an equilateral triangle.

Let's first consider the incircle of equilateral triangle ABC. The radius of the incircle, denoted as r, can be found using the formula:

r = (s √3) / 6

where s is the side length of the equilateral triangle. Since the circumradius of the triangle is given as 10, the side length can be found using the formula:

s = (2R) / √3

where R is the circumradius. Substituting the value of R = 10, we can find the value of s:

s = (2 * 10) / √3
s = 20 / √3

Now, let's consider circle Γ1, which is inscribed inside circle Γ. Since circle Γ1 is tangential to radii OC and OB, we can draw radii OC1 and OB1 from the center O to the points of tangency.

Now, let's consider the right triangle OOC1. The length of the hypotenuse OC1 is equal to the sum of the radii of circles Γ and Γ1:

OC1 = OC + C1C

Since OC is the circumradius of the equilateral triangle Γ, which is given as 10, we need to find C1C to determine OC1.

By symmetry, angles OOC1 and OCB1 are equal. Therefore, triangle OOC1 and triangle OCB1 are similar, making use of the angle-angle similarity criterion.

Using the property of an equilateral triangle, the angles C1OC1 and C1OB1 are equal to 120 degrees, and we know that angle COB is equal to 60 degrees. Thus, angle OCB1 is equal to half the difference between 120 and 60 degrees, or 30 degrees.

Since triangle OCB1 is a 30-60-90 triangle, we can use the length of OC to find C1C:

C1C = OC * (√3 / 2)
= 10 * (√3 / 2)
= 5√3

Now, we can use this value to find the radius of circle Γ1:

OC1 = OC + C1C
= 10 + 5√3

The radius of circle Γ1 is equal to OC1, so the radius of Γ1 can be expressed as 10 + 5√3.

To represent it in the requested form ab√(-c), we can multiply the expression by (√(-1)) / (√(-1)), which is equal to (√(-1)):

(10 + 5√3) * (√(-1)) / (√(-1))
= (√(-1)(10 + 5√3)) / √(-1)(√(-1))
= (10√(-1) + 5√3√(-1)) / (-1)
= (10√(-1) + 5√3√(-1)) / (-1)
= -(10√(-1) + 5√3√(-1))
= -(10√(-1) + 5√3√(-1))
= -10√(-1) - 5√3√(-1)
= -10i√1 - 5i√3√1
= -10i - 5i√3

So, the radius of Γ1 can be expressed as -10i - 5i√3.

From the given form, we can see that a = 10, b = 5, and c = 3.

Therefore, the value of a + b + c is 10 + 5 + 3 = 18.