Explanation:

To understand the ratio of the circumradius, inradius, and exradius of an equilateral triangle, let's define each of these terms and their relationships.

Circumradius:
The circumradius of a triangle is the radius of the circumcircle, which is the circle passing through all three vertices of the triangle.

Inradius:
The inradius of a triangle is the radius of the incircle, which is the circle tangent to all three sides of the triangle.

Exradius:
The exradius of a triangle is the radius of the excircle, which is the circle tangent to one side of the triangle and the extensions of the other two sides.

Now, let's consider an equilateral triangle with side length 'a'. To find the ratios, we need to determine the values of the circumradius, inradius, and exradius.

Circumradius:
In an equilateral triangle, the circumradius is the distance from the center of the triangle to any vertex. To calculate this, we can draw an altitude from any vertex to the midpoint of the opposite side, forming a right-angled triangle. The hypotenuse of this triangle is the circumradius.

By applying Pythagoras theorem, we can find that the length of the hypotenuse (circumradius) is (2/3)a. Thus, the circumradius is (2/3)a.

Inradius:
In an equilateral triangle, the inradius is the distance from the center of the triangle to any of its sides. To calculate this, we can draw an altitude from the center of the triangle to one of its sides, forming a right-angled triangle. The altitude is also the inradius.

By applying Pythagoras theorem, we can find that the length of the altitude (inradius) is (1/2)(√3)a. Thus, the inradius is (1/2)(√3)a.

Exradius:
In an equilateral triangle, each side is tangent to an excircle. The length of the exradius can be found by drawing an altitude from the center of the excircle to the side of the triangle, forming a right-angled triangle. The altitude is also the exradius.

By applying Pythagoras theorem, we can find that the length of the altitude (exradius) is (√3)a. Thus, the exradius is (√3)a.

Ratio Calculation:
Now, let's calculate the ratio of the circumradius, inradius, and exradius.

The ratio of the circumradius to the inradius is:
(2/3)a : (1/2)(√3)a
Simplifying this ratio, we get:
(4/3) : (√3/2)

Similarly, the ratio of the inradius to the exradius is:
(1/2)(√3)a : (√3)a
Simplifying this ratio, we get:
(1/2) : 1

Therefore, the overall ratio of the circumradius, inradius, and exradius is:
(4/3) : (√3/2) : (1/2) : 1

To simplify this ratio, we can multiply all terms by 6 to eliminate fractions:
8 : 3√3 : 3 :