\section*{Problem 1 - 071221}

Let \(x_{k}\) and \(y_{k}\) be integers that satisfy the conditions \(0 \leq x_{k} \leq 2\) and \(0 \leq y_{k} \leq 2\). a) Determine the number of all (non-degenerate) triangles with vertices \(P_{k}\left(x_{k} ; y_{k}\right)\), where \(x_{k}, y_{k}\) are the Cartesian coordinates of \(\mathrm{Pk}\)!

Note: Two triangles \(\Delta_{1}\) and \(\Delta_{2}\) are considered the same if and only if every vertex of \(\Delta_{1}\) is also a vertex of \(\Delta_{2}\).

b) Provide the area measurements of all these triangles!
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a) There are a total of 9 pairs \((x, y)\) of "rational" (= integer) numbers with \(0 \leq x \leq 2\) and \(0 \leq y \leq 2\), thus 9 points \(P(x, y)\) with integer coordinates \(x, y \in \{0,1,2\}\). There are \(\binom{9}{3}=84\) ways to choose 3 of these 9 points. However, in 8 cases, the 3 points lie on a straight line (either parallel to the \(\mathbf{x}\)- or \(\mathrm{y}\)-axis or on one of the diagonals \(y=x\) or \(y=3-x\)). Therefore, in \(84-8=76\) cases, the three points form a non-degenerate triangle.

b) There are 8 classes of triangles, where the triangles of one class are congruent to each other. Namely:

\begin{tabular}{clcc} 
Class & Representative & Number & Area \\
1. & \((0,0),(0,1),(1,0)\) & 16 & \(1 / 2\) \\
2. & \((0,0),(0,1),(1,2)\) & 16 & \(1 / 2\) \\
3. & \((0,0),(0,1),(2,0)\) & 16 & 1 \\
4. & \((0,0),(0,1),(2,2)\) & 8 & 1 \\
5. & \((0,0),(0,2),(1,2)\) & 8 & 1 \\
6. & \((0,0),(0,2),(2,0)\) & 4 & 2 \\
7. & \((0,0),(0,2),(2,1)\) & 4 & 2 \\
8. & \((0,0),(1,2),(2,1)\) & 4 & \(3 / 2\) \\
Sum: & & 76 &
\end{tabular}