Suppose $(x_i,y_i),$ $i=1,2,3,4$ are the vertices of a convex quadrilateral, in order as you follow the boundary (so that $i=1,3$ are opposite points as are $i=2,4.$ I wanted to find the center of gravity of the interior of the quadrilateral.
For a triangle it's the same as the center of gravity of the three vertices, so split the quadrilateral into two triangles in two different ways: one in which the two triangles correspond to the indices $1,2,3$ and $3,4,1,$ and one in which they correspond to $2,3,4$ and $4,1,2.$ The point we seek should be on the line between the first two centers of gravity, and also on the line between the next two. Thus \begin{align} & r\cdot\frac{x_1+x_2+x_3} 3 + (1-r)\cdot\frac{x_3+x_4+x_1} 3 \\[8pt] = {} & x\text{-coordinate of the center of gravity} \\[8pt] = {} & s\cdot\frac{x_2+x_3 + x_4} 3 + (1-s)\cdot \frac{x_4+x_1+x_2} 3. \end{align} and then a second equation results from replacement of every $x_i$ by $y_i,$ giving the $y$-coordinate.
Solving for $r$ and $s$ I get \begin{align} r & = \frac{(x_3-x_1)(y_2-y_3)-(y_3-y_1)(x_2-x_3)}{(x_3-x_1)(y_2-y_4) - (y_3-y_1)(x_2-x_4)}, \\[8pt] 1-r & = \frac{(x_3-x_1)(y_3-y_4) - (y_3-y_1)(x_3-x_4)}{(x_3-x_1)(y_2-y_4) - (y_3-y_1)(x_2-x_4)}. \end{align} Hence \begin{align} & x\text{-coordinate of the center of gravity} \\[8pt] = {} & r\cdot\frac{x_1+x_2+x_3} 3 + (1-r)\cdot\frac{x_3+x_4+x_1} 3 \\[8pt] = {} & \frac{x_1+x_2+x_3+x_4} 3 - \frac 1 3 \cdot \frac{(x_3-x_1)(x_4y_2 - y_4x_2) - (x_2-x_4)(x_1y_3 - x_3 y_1)}{(x_3-x_1)(y_2-y_4) - (y_3-y_1)(x_2-x_4)} \\[8pt] = {} & \cdots\text{etc.}\cdots \end{align} and so now we should massage it into some form that has some nice symmetries and simplicity (in particular, readily seen to be invariant under cyclic shifts of the indices $1,2,3,4$), BUT at this point I'm thinking this must be in some standard tables and books somewhere. So that is my question: Where is this in the literature?
What you did is essentially explained by Möbius (see p.211-215, section 113), where on page 212, he manipulated with (1), (2) and (3) similar to your formulas, except for different use of coordinates. See also this.