Question 161
Suppose $a$, $b$, $c$ are the side lengths of a triangle $ABC$. Prove that the system of equations
$$
\begin{aligned}
by + cz + aw &= 1, \\
bx + az + cw &= 0, \\
cx + ay + bw &= 0, \\
ax + cy + bz &= 0,
\end{aligned}
$$
has a unique solution.
$$
\begin{aligned}
by + cz + aw &= 1, \\
bx + az + cw &= 0, \\
cx + ay + bw &= 0, \\
ax + cy + bz &= 0,
\end{aligned}
$$
has a unique solution.