4. (10 points) On a $9 \times 9$ grid paper, the vertices of $1 \times 1$ small squares are called grid points. As shown in the figure, the three vertices of triangle $ABC$ are all grid points. If a grid point $P$ makes the areas of triangle $PAB$ and triangle $PAC$ equal, then point $P$ is called a "good point". How many "good points" are there on this grid paper? $\qquad$
【Analysis】As shown in the figure below, draw a line through point $A$ and the midpoint of side $B C$, intersecting the grid at points $E$ and $F$. It can be proven that points $D$, $E$, and $F$ are all good points;

Draw a parallel line through points $A$ and $B$, forming a parallelogram with the original triangle, resulting in parallelogram $A C B I$. It can be proven that points $I$, $H$, and $G$ are also good points.
【Solution】
(1) $\triangle B D A$ and $\triangle C D A$ have the same base and height, so their areas are equal;
(2) The areas of $\triangle A B E$ and $\triangle A C E$ are both half the area of parallelogram $A B C E$, so their areas are equal;
(3) The area of $\triangle A B F$ = the area of $\triangle B D F$ - the area of $\triangle B D A$, and the area of $\triangle C A F$ = the area of $\triangle C D F$ - the area of $\triangle C D A$. Since the areas of $\triangle B D A$ and $\triangle C D A$ are equal, the area of $\triangle A B F$ = the area of $\triangle C A F$;
(4) The areas of $\triangle A B I$ and $\triangle A C I$ are both half the area of parallelogram $A C B I$, so they are equal;
(5) The area of $\triangle A B H$ is half the area of $\triangle A B I$, and the area of $\triangle A C H$ is half the area of $\triangle A C I$, so the areas of $\triangle A B H$ and $\triangle A C H$ are equal;
(6) $\triangle A G B$ and $\triangle A G C$ have the same base $A G$, and the heights from this base to the two triangles are equal, so the areas of these two triangles are equal.

Therefore, there are a total of 6 good points in this problem.