Problem 996

a) A line passing through the points $(0,0)$ and $(x_{1}, y_{1})$ is given by the equation $y_{1} x - x_{1} y = 0$. Therefore, according to problem 12.75 B, the distance from the point $(x_{2}, y_{2})$ to this line is $\frac{\left|y_{1} x_{2} - x_{1} y_{2}\right|}{\sqrt{x_{1}^{2} + y_{1}^{2}}}$. This distance is equal to the height of the considered triangle, dropped to the side of length $\sqrt{x_{1}^{2} + y_{1}^{2}}$.

b) The area of the considered triangle is equal to the area of the triangle with vertices at points $(0,0)$, $(x_{1} - x_{3}, y_{1} - y_{3})$, and $(x_{2} - x_{3}, y_{2} - y_{3})$. Using the formula from part a), we obtain the required result.