Example 6. Let the three sides of $\triangle A B C$ be $a, b, c$ with opposite angles $A, B, C$, respectively, and satisfy the condition $a \cos A+b \cos B=c \cos C$. Then $\triangle A B C$ must be ( ).
(A) an equilateral triangle
(B) a right triangle with $a$ as the hypotenuse
(C) a right triangle with $b$ as the hypotenuse
(D) none of the above (A), (B), (C) is correct.
This is a problem from the 1983 Shanghai Junior High School Mathematics Competition, and book [7] uses it as an example of the application of the cosine rule, converting the condition in the problem into a relationship between the three sides using the cosine rule, and then simplifying it to make a judgment.
Answer the multiple-choice question, just get it right, no need to argue. Therefore, it is valuable to be quick and accurate. From the condition
$$
a \cos A + r \cos B = c \cos C
$$

it is easy to see that $A(a), [B(b)]$ and $C(c)$ cannot be interchanged, so we know that conclusion (A) is incorrect.

Since the above equation is symmetric with respect to $A(a)$ and $B(b)$, we also know that conclusion (B) $\Leftrightarrow$ conclusion (C). Neither of them is correct.
Therefore, the answer should be (D).
Of course, using the cosine rule, we can derive a more explicit conclusion $a^2 = b^2 + c^2$ or $b^2 = a^2 + c^2$,

which means $\triangle ABC$ must be a right triangle, with either $a$ or $b$ as the hypotenuse.