If points $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right), P_{3}\left(x_{3}, y_{3}\right)$ are on the curve $x y=1$, then the area of triangle $P_{1} P_{2} P_{3}$ is:

$$
t=\frac{\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)\left(x_{3}-x_{1}\right)}{2 x_{1} x_{2} x_{3}}
$$
It is known that the area of the triangle $P_{1} P_{2} P_{3}$ is:

$$
t=\frac{1}{2}\left|\begin{array}{lll}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{array}\right|
$$

From the equation

$$
x y=1
$$

we have

$$
y_{i}=\frac{1}{x_{i}} \quad(i=1,2,3)
$$

thus:

$$
t=\frac{1}{2}\left|\begin{array}{lll}
x_{1} & \frac{1}{x_{1}} & 1 \\
x_{2} & \frac{1}{x_{2}} & 1 \\
x_{3} & \frac{1}{x_{3}} & 1
\end{array}\right|=\frac{\left|\begin{array}{lll}
x_{1}^{2} & 1 & x_{1} \\
x_{2}^{2} & 1 & x_{2} \\
x_{3}^{2} & 1 & x_{3}
\end{array}\right|}{2 x_{1} x_{2} x_{3}}=\frac{\left|\begin{array}{ccc}
1 & x_{1} & x_{1}^{2} \\
1 & x_{2} & x_{2}^{2} \\
1 & x_{3} & x_{3}^{2}
\end{array}\right|}{2 x_{1} x_{2} x_{3}}
$$

Subtracting the last row from the first and second rows of the determinant:

$$
\begin{aligned}
& t=\frac{\left|\begin{array}{ccc}
0 & x_{1}-x_{3} & x_{1}^{2}-x_{3}^{2} \\
0 & x_{2}-x_{3} & x_{2}^{2}-x_{3}^{2} \\
1 & x_{3} & x_{3}^{2}
\end{array}\right|}{2 x_{1} x_{2} x_{3}}=\frac{\left(x_{1}-x_{3}\right)\left(x_{2}-x_{3}\right)}{2 x_{1} x_{2} x_{3}}\left|\begin{array}{ccc}
0 & 1 & x_{1}+x_{3} \\
0 & 1 & x_{2}+x_{3} \\
1 & x_{3} & x_{3}^{2}
\end{array}\right|= \\
& =\frac{\left(x_{1}-x_{3}\right)\left(x_{2}-x_{3}\right)}{2 x_{1} x_{2} x_{3}}\left|\begin{array}{ccc}
0 & 0 & x_{1}-x_{3} \\
0 & 1 & x_{2}+x_{3} \\
1 & x_{3} & x_{3}^{2}
\end{array}\right|=\frac{\left(x_{1}-x_{3}\right)\left(x_{2}-x_{3}\right)\left(x_{3}-x_{1}\right)}{2 x_{1} x_{2} x_{3}}
\end{aligned}
$$

(Adolf Szűcs, Budapest.)

The problem was also solved by: Ádámffy E., Bartók I., Biró A., Brambring V., Braun I., Dömény E., Dömény I., Enyedi B, Haar A., Hausvater I., Kertész G., Kiss J., Kürti I., Liebner A., Pichler S., Pivnyik I., Rássy P., Riesz K., Rosenberg J., Schöffer I., Schwemmer I., Sonnenfeld J.