The Trapezoid Method is a general technique for calculating the area of any simple polygon (in this case, a triangle) in a plane. The core idea is to project each vertex of the polygon onto a coordinate axis (e.g., the x-axis), forming a series of trapezoids between the polygon's edges and the axis.
By calculating the algebraic sum (i.e., a sum that includes both positive and negative values) of the areas of these trapezoids, one can accurately determine the area of the original shape. The diagram below demonstrates this process: the area of triangle ABC is found by adding the areas of the two green trapezoids and then subtracting the area of the red trapezoid.
The area of a trapezoid formed by two points $(x_1, y_1)$ and $(x_2, y_2)$ and their projections on the x-axis can be expressed as: $$ \text{Area}_{12} = \frac{(y_1 + y_2) \times (x_1 - x_2)}{2} $$
Here we use $(x_1 - x_2)$ instead of its absolute value because its sign automatically handles whether the area should be added or subtracted, depending on the order of the points (clockwise or counter-clockwise). This makes the formula extremely versatile.
Therefore, the area of triangle ABC is the absolute value of the algebraic sum of the areas of the trapezoids corresponding to its three sides: $$ \text{Area}(\triangle ABC) = \left| \frac{(y_A+y_B)(x_A-x_B)}{2} + \frac{(y_B+y_C)(x_B-x_C)}{2} + \frac{(y_C+y_A)(x_C-x_A)}{2} \right| $$
Interestingly, if you expand and simplify this formula, you will find that its algebraic form is identical to the Shoelace Formula. This demonstrates that the geometric meaning of the Shoelace Formula is derived from this very concept.
An Apollonian Circle is the set of all points P in a plane such that the ratio of the distances from P to two fixed points, A and B, is a constant value k. That is, $PA/PB = k$.
This diagram shows the case where $A=(4,0)$, $B=(1,0)$, and the ratio $k=2$.