Writing a probability in terms of the edges of a simplex I describe the problem dimension $2$, but it could be generalized to $n$ dimensions.
So we have $X_{1}, X_{2}, X_{3}$ three $iid$ random variables of continuous law $F(x,y)$ on $\Bbb R^2$. Let's denote by $S[X_{1}, X_{2}, X_{3}]$ the simplex generated by those random variables. Given a point $(x,y)$, is it possible to write $P((x,y)\in S[X_{1}, X_{2}, X_{3}])$ as a function $g$ of $F(x,y)$ ? For example, in dimension $1$, $P(x\in [X_{1}, X_{2}]) = 2F(x)(1-F(x))$.
Thank you

This is based on the barycentric coordinates. Applying Cramer's rule or using suitable software, one obtains
\begin{aligned}
a_1 &= \frac{x_2 y_3 - y_2 x_3 + x_3 y - x y_3 - x_2 y + y_2 x}{-x_2 y_1 + x_2 y_3 - x_1 y_3 + y_1 x_3 + y_2 x_1 - y_2 x_3} \, , \\
\\
a_2 &= \frac{x_1 y - x_1 y_3 - x_3 y + y_1 x_3 + x y_3 - y_1 x}{-x_2 y_1 + x_2 y_3 - x_1 y_3 + y_1 x_3 + y_2 x_1 - y_2 x_3} \, , \\
\\
a_3 &= \frac{-x_2 y_1 - x_1 y + y_1 x + x_2 y + y_2 x_1 - y_2 x}{-x_2 y_1 + x_2 y_3 -x_1 y_3 + y_1 x_3 + y_2 x_1 - y_2 x_3} \, ,
\end{aligned}
where $X_i=(x_i,y_i)$ and $X=(x,y)$. It does not look like there is a simple rule to find the law of $a_i$, but the denominators in $\lbrace a_1, a_2, a_3\rbrace$ are all the same, and the numerators are very similar. One may view the problem with a different point of view. Indeed, one can compute the edge lines of the triangle:
\begin{aligned}
E_{12}: & \;\left(y - y_1\right) \left(x_2 - x_1\right) = \left(y_2 - y_1\right) \left(x - x_1\right) \, , \\
E_{13}: & \;\left(y - y_1\right) \left(x_3 - x_1\right) = \left(y_3 - y_1\right) \left(x - x_1\right) \, , \\
E_{23}: & \;\left(y- y_2\right) \left(x_3 - x_2\right) = \left(y_3 - y_2\right) \left(x - x_2\right) \, . \\
\end{aligned}
The point $X=(x,y)$ belongs to the triangle if $X$ belongs to the same half-plane as $X_3$ with respect to $E_{12}$:
\begin{aligned}
\text{sign}\left(\left(y - y_1\right) \left(x_2 - x_1\right) - \left(y_2 - y_1\right) \left(x - x_1\right)\right) = \text{sign}\left(\left(y_3 - y_1\right) \left(x_2 - x_1\right) - \left(y_2 - y_1\right) \left(x_3 - x_1\right)\right) ,
\end{aligned}
etc. The latter equation is not very different from $a_3\geq 0$.
