In coordinate geometry, the area of a triangle formed by three points can be calculated using determinants, vector products, or coordinate formulas.
Determinant form:
\[ A = \frac{1}{2} \left| \begin{vmatrix} x_2 - x_1 & y_2 - y_1 \\ x_3 - x_1 & y_3 - y_1 \end{vmatrix} \right| \]
Expanded version:
\[ A = \frac{1}{2} \left[ (x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1) \right] \]
Alternate symmetric form:
\[ A = \frac{1}{2} \left[ x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right] \]
\[ A = \frac{1}{2} \left| x_1 y_2 - x_2 y_1 \right| \]
This formula simplifies computation when one vertex is at the origin.
