Task 500
Task 500: test task in emrekuruu/math-openenv, an RL environment on the Hugging Face Hub. Let $x 1,$ $x 2,$ $x 3,$ $y 1,$ $y 2,$ and $y 3$ be real numbers such that \begin{align } (x 1 - x 2)^2 + (y 1 - y 2)^2 &= 9, \\ (x 1 - x 3)^2 + (y 1 - y 3)^2 &= 16, \\ (x 2 - x 3)^2 + (y 2 - y 3)^2 &= 25…
The task
Let $x_1,$ $x_2,$ $x_3,$ $y_1,$ $y_2,$ and $y_3$ be real numbers such that \begin{align*} (x_1 - x_2)^2 + (y_1 - y_2)^2 &= 9, \\ (x_1 - x_3)^2 + (y_1 - y_3)^2 &= 16, \\ (x_2 - x_3)^2 + (y_2 - y_3)^2 &= 25. \end{align*}Find $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}^2.$