Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be [i]good [/i]if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ [i]smaller triangles of equal area.[/i] Determine the number of good points for a given triangle $ABC$.
To determine the number of good points \( P \) in a triangle \( ABC \) such that \( P \) can be used to divide the triangle into 27 smaller triangles of equal area using 27 rays emanating from \( P \), we need to follow these steps:

1. **Understanding the Problem:**
   - We need to divide the triangle \( ABC \) into 27 smaller triangles of equal area.
   - This requires 27 rays emanating from point \( P \).

2. **Initial Observations:**
   - Three of these rays must be \( PA, PB, \) and \( PC \), which connect \( P \) to the vertices of the triangle.
   - This leaves us with 24 additional rays to be distributed among the sides of the triangle.

3. **Distribution of Rays:**
   - To divide the triangle into 27 smaller triangles of equal area, each side of the triangle must be divided into equal segments.
   - Let’s denote the points where the rays intersect the sides of the triangle as "great points."

4. **Equal Segments:**
   - Since we need 27 smaller triangles, and each smaller triangle must have equal area, the sides of the triangle must be divided into equal segments.
   - Let’s consider the side \( AB \). If we place 6 great points on \( AB \), it will divide \( AB \) into 7 equal segments.
   - Similarly, we need to place 6 great points on \( BC \) and 6 great points on \( CA \).

5. **Verification:**
   - Placing 6 great points on each side results in 7 segments per side.
   - The rays from \( P \) to these points will create 27 smaller triangles of equal area.

6. **Conclusion:**
   - The only point \( P \) that can achieve this division is the centroid of the triangle \( ABC \).
   - The centroid is the point where the medians of the triangle intersect, and it divides each median into a ratio of 2:1.

Therefore, the number of good points \( P \) in a given triangle \( ABC \) is exactly one, which is the centroid of the triangle.

\(\blacksquare\)

The final answer is \( \boxed{1} \)