2. Given that $A D$ is the altitude of $\triangle A B C$ (inside the shape). The following four conditions are given:
(1) $A B+B D=A C+C D$;
(2) $A B-B D=A C-C D$;
(3) $A B \cdot B D=A C \cdot C D$;
(4) $A B: B D=A C: C D$.
The number of conditions that can definitely lead to $A B=A C$ is ( ) .
(A) 1
(B) 2
(C.) 3
(D) 4
2. D.

As shown in Figure 2, let
$$
\begin{array}{l}
A B=c, A C=b, \\
B D=x, C D=y .
\end{array}
$$

Then $c^{2}-x^{2}$
$$
=b^{2}-y^{2} \text {. }
$$

Factoring gives
$$
\begin{array}{l}
(c+x)(c-x) \\
=(b+y)(b-y) .
\end{array}
$$
(1) If $c+x=b+y$, then $c-x=b-y$.

Adding the two equations yields $2 c=2 b \Rightarrow c=b$.
(2) If $c-x=b-y$, similarly, $c=b$.
(3) If $c x=b y$, then $y=\frac{c x}{b}$, substituting into equation (1) gives
$$
\begin{array}{l}
c^{2}-b^{2}=x^{2}\left(1-\frac{c^{2}}{b^{2}}\right) \\
\Rightarrow(c+b)(c-b)\left(b^{2}+x^{2}\right)=0 \Rightarrow c=b .
\end{array}
$$
(4) If $c: x=b: y$, then $y=\frac{b x}{c}$, substituting into equation (1) similarly yields $c=b$.
In summary, all four conditions lead to $A B=A C$.