4. Given that $A D$ is the internal angle bisector of $\triangle A B C$, and the extension of $A D$ intersects the circumcircle of $\triangle A B C$ at point $E$. Then the relationship between $A B \cdot A C - B D \cdot D C$ and $A D^{2}$ is $\qquad$ (fill in “equal” or “not equal”).


4. Equality.

As shown in Figure 7, connect $B E$.
Since $\angle B A E=\angle D A C$,
$$
\angle E=\angle C \text {, }
$$

we have $\triangle A B E \backsim \triangle A D C$.
Therefore, $\frac{A B}{A D}=\frac{A E}{A C}$, which means
$$
A B \cdot A C=A E \cdot A D \text {. }
$$

By the intersecting chords theorem, we get
$$
B D \cdot D C=D E \cdot A D \text {. }
$$
(1) - (2) yields
$$
A B \cdot A C-B D \cdot D C=(A E-D E) \cdot A D=A D^{2} \text {. }
$$