# Generate LaTex code for making a graph joining points (1, -1), (-0.5, 0.5) and (1, 2) which makes a triangle. In the code, also give the solution to show that the triangle formed is isosceles.

To create a graph of the triangle formed by the points \((1, -1)\), \((-0.5, 0.5)\), and \((1, 2)\) and to prove that it is isosceles, you can use the following LaTeX code with the `tikz` package for the graph and some mathematical reasoning for the proof.
```latex
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\begin{document}
\begin{center}
% Drawing the graph
\begin{tikzpicture}[scale=1.5]
% Draw axes
\draw[->] (-2,0) -- (2,0) node[anchor=north] {x};
\draw[->] (0,-2) -- (0,3) node[anchor=east] {y};
% Draw the triangle
\draw[thick] (1, -1) -- (-0.5, 0.5) -- (1, 2) -- cycle;
% Mark the points
\filldraw[black] (1, -1) circle (1pt) node[anchor=east] {$(1, -1)$};
\filldraw[black] (-0.5, 0.5) circle (1pt) node[anchor=south east] {$(-0.5, 0.5)$};
\filldraw[black] (1, 2) circle (1pt) node[anchor=west] {$(1, 2)$};
% Draw the right angle
% \draw[fill=white] (-0.5, 0.5) circle (1pt);
\end{tikzpicture}
\end{center}
\section*{Proving the triangle is isosceles}
To prove that the triangle with vertices \( A(1, -1) \), \( B(-0.5, 0.5) \), and \( C(1, 2) \) is isosceles, we need to calculate the lengths of the sides \( AB \), \( AC \), and \( BC \).
The length of a line segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
1. **Calculate \( AB \)**:
\[
AB = \sqrt{((-0.5) - 1)^2 + (0.5 - (-1))^2} = \sqrt{(-1.5)^2 + (1.5)^2} = \sqrt{2.25 + 2.25} = \sqrt{4.5} = \frac{3\sqrt{2}}{2}
\]
2. **Calculate \( AC \)**:
\[
AC = \sqrt{(1 - 1)^2 + (2 - (-1))^2} = \sqrt{0^2 + (3)^2} = \sqrt{9} = 3
\]
3. **Calculate \( BC \)**:
\[
BC = \sqrt{(1 - (-0.5))^2 + (2 - 0.5)^2} = \sqrt{(1.5)^2 + (1.5)^2} = \sqrt{2.25 + 2.25} = \sqrt{4.5} = \frac{3\sqrt{2}}{2}
\]
Now we can compare the lengths:
- \( AB = \frac{3\sqrt{2}}{2} \)
- \( AC = 3 \)
- \( BC = \frac{3\sqrt{2}}{2} \)
Since \( AB = BC \), the triangle is isosceles.
\end{document}
```
This code creates a triangle defined by the specified points and includes calculations to show that two sides of the triangle are equal, confirming that it is isosceles. To compile this code, use a LaTeX editor that supports TikZ, such as Overleaf or TeXShop.