3) A non-degenerate triangle is given whose sides have lengths of three natural numbers $a, b$, and $c$. Knowing that $b$ and $c$ are multiples of $a$, what can we say about the triangle?

(A) It always has an integer area

(B) it always has a rational area, but not necessarily an integer

(C) it is necessarily isosceles

(D) it could be a right triangle

(E) no such triangles exist.
3) The answer is (C). Let $\beta=\frac{b}{a}$ and $\gamma=\frac{c}{a}$. Since $b$ and $c$ are multiples of $a$, $\beta$ and $\gamma$ are integers. We write the triangle inequality relative to side $a$:

$$
a>|c-b|=|a \gamma-a \beta|=a \cdot|\gamma-\beta|
$$

and thus, simplifying $a$, we get

$$
1>|\gamma-\beta|
$$

and since $\gamma$ and $\beta$ are integers, it must be that $|\gamma-\beta|=0$, which implies $b=c$. Therefore, the triangle must be isosceles: the answer (C) is correct. A counterexample to answers (A), (B), and (E) is provided by the equilateral triangle with side length 1. Using the Pythagorean theorem, it can also be shown that answer (D) is incorrect.