Let $ABCD$ be a square with side length $1$. We are tasked to determine the number of points $P$ inside the square such that the square can be partitioned into $10$ triangles of equal area, all having $P$ as a common vertex.

To solve this problem, consider the following steps:

1. Understanding the Equal Area Condition: For the square to be divided into 10 triangles of equal area, each triangle must have an area equal to $\frac{1}{10}$ because the total area of the square is $1$.

2. Forming Triangles: Each of the triangles must share vertex $P$. Thus, $P$ serves as a vertex to all 10 triangles.

3. Geometric Consideration: Consider an arbitrary point $P$ in the interior of the square. For $P$ to be a common vertex to triangles of equal area, it must be connected to the vertices of the square or points along its perimeter in such a way that results in equal partitioning.

4. Central Symmetry and Regular Division: By symmetry and the nature of equal division, the intersection points of lines radiating from $P$ to the sides and vertices of the square should ideally divide the sides or regions into segments that are proportional and compatible with creating triangles of equal area.

5. **Specific Positioning of $P$**: The lines radiating from $P$ to the vertices and sides of the square should be symmetric. The regularity condition can be satisfied by placing $P$ at positions towards the center with multiplicity in terms of symmetry.

6. **Counting Suitable Positions for $P$**: By solving these conditions systematically or employing symmetry arguments:
- Consider dividing the square into 4 equal smaller squares. The center of each of these smaller squares can potentially serve a suitable point $P$.
- Each smaller square has 4 quadrants (formed by diagonals and mid-segments), which when combined with the central symmetry provided by the square, can lead to potential points.

Consequently, there are $4 \times 4 = 16$ suitable locations for $P$ based on symmetry and the layout described.

Thus, the number of points $P$ such that the square can be divided into 10 triangles of equal area with $P$ as a vertex is:
$$\boxed{16}.$$