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?

Question

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?

Answer 1

To find the radius of circle Γ1, we need to break down the problem into smaller steps.

Step 1: Finding the length of side AB of the equilateral triangle ABC:
Since ABC is an equilateral triangle, all sides are equal. Let's assume the length of side AB is "x". If we draw perpendiculars from O to sides AB, BC, and AC, we get three congruent right-angled triangles, with hypotenuse of 10 (circumradius) and one leg of length x/2. Using the Pythagorean theorem, we can find the length of the other leg:

(10)^2 = (x/2)^2 + x^2
100 = x^2/4 + x^2
100 = 5x^2/4
x^2 = (100*4)/5 = 80
x = √80 = 4√5

So, the length of side AB is 4√5 units.

Step 2: Finding the radius of Γ1:
Since Γ1 is tangential to radii OC and OB, we can draw radii from O to the points of tangency on circle Γ1. These radii are perpendicular to the sides of triangle ABC. Let's assume the radius of Γ1 is "r". Join the center O of circle Γ to the incenter I of circle Γ1. This line will pass through the points of tangency of circle Γ1 with sides AB, BC, and AC.

Remember that the incenter of an equilateral triangle is also the circumcenter, and it is located at the intersection point of the three angle bisectors. Therefore, the line joining O to I of circle Γ1 will pass through the center O of Γ and divide it into two equal parts (since it is an equilateral triangle).

Let's consider the right-angled triangle OBI. The hypotenuse is 10 (circumradius), and one leg is r (radius of Γ1). The other leg can be found using Pythagorean theorem:

(10)^2 = r^2 + r^2
100 = 2r^2
r^2 = 100/2 = 50

The radius of circle Γ1 is √50 = 5√2.

Step 3: Finding a, b, and c:
The radius of Γ1 can be expressed as ab√(−c). In this case, a = 5, b = 2, and c = 2.

Therefore, the value of a + b + c is 5 + 2 + 2 = 9.

So, the value of a + b + c is 9.