The equations of the sides of a quadrilateral are:

$$
y=-x+7, \quad y=\frac{x}{2}+1, \quad y=-\frac{3}{2} x+21, \quad y=\frac{7}{4} x+\frac{3}{2}
$$

Determine the coordinates of the vertices of the quadrilateral and its area.
If the sides of the quadrilateral $A B C D$ are $a, b, c, d$, then the equations of these lines are:

$$
a \equiv y=-x+7
$$

$$
b \equiv y=\frac{x}{2}+1
$$

$$
c \equiv y=-\frac{3}{2} x+21
$$

$$
d \equiv y=\frac{7}{4} x+\frac{3}{2}
$$

The coordinates of point $A$ are obtained from equations (1) and (4):

$$
A \ldots \xi_{1}=2 ; \quad \eta_{1}=5
$$

The coordinates of $B$ are given by equations (1) and (2):

$$
\xi_{2}=4 ; \quad \eta_{2}=3
$$

The coordinates of $C$ are given by equations (2) and (3):

$$
\xi_{3}=10 ; \quad \eta_{3}=6
$$

and finally, the coordinates of $D$ are from equations (3) and (4):

$$
\xi_{4}=6 ; \quad \eta_{4}=12
$$

The area of the quadrilateral:

$$
\begin{gathered}
T_{A B C D \square}=T_{A B C \triangle}+T_{C D A \triangle}= \\
=\frac{1}{2}\left|\begin{array}{ccc}
\xi_{1} & \xi_{2} & \xi_{3} \\
\eta_{1} & \eta_{2} & \eta_{3} \\
1 & 1 & 1
\end{array}\right|+\frac{1}{2}\left|\begin{array}{ccc}
\xi_{3} & \xi_{4} & \xi_{1} \\
\eta_{3} & \eta_{4} & \eta_{2} \\
1 & 1 & 1
\end{array}\right|=\frac{1}{2}\left[\left(\xi_{1}-\xi_{3}\right)\left(\eta_{2}-\eta_{4}\right)+\left(\xi_{2}-\xi_{4}\right)\left(\eta_{3}-\eta_{1}\right)\right]=35
\end{gathered}
$$

square units.

(János Szombathy, Fiume.)

The problem was also solved by: E. Ádámffy, I. Bartók, I. Braun, I. Dömény, E. Dömény, B. Enyedi, A. Haar, Z. Harsányi, I. Kürti, G. Kerlész, J. Kiss, A. Liebner, M. Pám, S. Pichler, I. Pivnyik, P. Rássy, K. Riesz, J. Rosenberg, A. Szúcs, I. Schwemmer, I. Székely.