Suppose B is a randomly scattered point in the segment [OM] as shown in Fig. System Model. The PDF of x (i.e., the distance between O and B) is known as $f_{X}$, and all the other points (A, C, M, and O) are fixed as described in the figure, i.e., $H_u$, $H_b$, and $r$ are deterministic.
I look forward to deriving the distribution of $d_1+d_2$, i.e., the distance sum between A,B, and C. I have tried to derive the distribution by inverting the following expression w.r.t $x$, but the result gives a polynomial of fourth degree that is unfeasible to simplify \begin{align} &\mathbb{P}\left(d_1+d_2<u\right)=\mathbb{P}\left(\sqrt{x^2+H_u^2}+\sqrt{\left(\left(H_u-H_b\right)^2+\left(r-x\right)^2\right)}<u\right)\\ &=\mathbb{P}\left(g_1(r,H_u,H_b)x^4+g_2(r,H_u,H_b)x^2+g_3(r,H_u,H_b)x+g_4(r,H_u,H_b)<g_5(r,H_u,H_b,u)\right) \end{align} Thank you very much for any suggestions or ideas!