342. Solve the system:

$$
\left\{\begin{array}{l}
x+y=a \\
y+z=b \\
z+x=c
\end{array}\right.
$$

Interpret the roots of this system geometrically, assuming $a, b, c$ are the sides of a triangle.
342. Adding the first two equations gives:

$$
x+2 y+z=a+b
$$

but

$$
x+z=c
$$

therefore,

$$
y=\frac{a+b-c}{2}
$$

Similarly:

$$
x=\frac{a+c-b}{2} ; z=\frac{b+c-a}{2}
$$

Benedetti also provides a geometric interpretation: consider a triangle with sides $a, b, c$ (Fig. 77). If a circle is inscribed in it, then at the points of tangency, the sides will be divided into segments such that $x+y=a ; \quad y+z=b$;

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-154.jpg?height=394&width=540&top_left_y=654&top_left_x=1158)

Fig. 77. $z+x=c$.