An equilateral triangle ABC is inscribed in a circle of radius 20√3 cm. The centroid of triangle ABC lies at a distance d from vertex A. What is the value of d (in cm)?

Introduction / Context:
This geometry question connects several important centers of an equilateral triangle: centroid, circumcenter and incenter. In an equilateral triangle, all these centers coincide at the same point. The problem asks for the distance from a vertex to the centroid, given the radius of the circumcircle.

Given Data / Assumptions:

• Triangle ABC is equilateral.
• Triangle ABC is inscribed in a circle (circumcircle) of radius 20√3 cm.
• The centroid G of triangle ABC is the same point as the circumcenter and incenter, because the triangle is equilateral.
• We must find the distance d from vertex A to point G.

Concept / Approach:
In any triangle, the centroid is the intersection of medians. In an equilateral triangle, there is a special symmetry: centroid, circumcenter, orthocenter and incenter all coincide at the same point. The distance from a vertex to the circumcenter is exactly the circumradius R. Therefore, once we know that G is also the circumcenter, the distance AG is simply R.

Step-by-Step Solution:
Step 1: Recognize that in an equilateral triangle, centroid, circumcenter and incenter coincide. Step 2: The circle of radius 20√3 cm is the circumcircle of triangle ABC. Step 3: The circumcenter is the center of this circle and is at distance R from each vertex. Step 4: Since centroid and circumcenter are the same point G, the distance from vertex A to G is equal to the circumradius R. Step 5: Therefore, d = AG = 20√3 cm.

Verification / Alternative check:
We can also use the formula for the distance from a vertex to centroid in an equilateral triangle. If a is the side length and h is the height, then the centroid divides each median in the ratio 2:1 from the vertex. The height is h = (√3 / 2) * a, and the centroid lies at distance 2h / 3 from a vertex. For an equilateral triangle with circumradius R, we also have R = a / √3. Combining these gives distance from vertex to centroid as a / √3 = R. This confirms our reasoning that AG equals R, which is 20√3 cm.

Why Other Options Are Wrong:
15 cm and 20 cm: Both are much smaller than the given radius and do not match the relation between centroid and circumcenter for an equilateral triangle.
30√3 cm: This is larger than the circumradius and would place the centroid outside the circle, which is impossible.
10√3 cm: This is half the radius, which would be the distance for some different construction, but not for centroid in an equilateral triangle.

Common Pitfalls:
A frequent mistake is to confuse centroid with the center of the incircle or to think that the centroid lies two thirds of the way from the vertex to the circumcenter, instead of along the median. In an equilateral triangle this confusion disappears if you remember that all the major centers coincide. Keeping this symmetry in mind avoids unnecessary calculations.

Final Answer:
The distance from vertex A to the centroid is 20√3 cm.