Given the spherical coordintes of point \( P_1(\rho_1,\theta_1,\phi_1) \) and point \( P_2(\rho_2,\theta_2,\phi_2) \), we first convert the coordinates of each point to rectangular coordinates written as \( P_1(x_1,y_1,z_1) \) and \( P_2(x_2,y_2,z_2) \)

where

\( x_1 = \rho_1 \sin \phi_1 \cos \theta_1 \) , \( y_1 = \rho_1 \sin \phi_1 \sin \theta_1 \) , \( z_1= \rho_1 \cos \phi_1 \)

\( x_2 = \rho_2 \sin \phi_2 \cos \theta_2 \) , \( y_2 = \rho_2 \sin \phi_2 \sin \theta_2 \) , \( z_2= \rho_2 \cos \phi_2 \)

The distance \( d(P_1 P_2) \) between points \( P_1 \) and \(P_2\) is given by

\( d(P_1 P_2) = \sqrt {(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2} \)

The rectangular coordinates of the midpoint \( M(x,y,z) \) of the segment \( P_1 P_2 \) are given by

\( x = \dfrac{x_1+x_2}{2} \) , \( y = \dfrac{y_1+y_2}{2} \) , \( z = \dfrac{z_1+z_2}{2} \)

Then the rectangular coordinates of the midpoint are converted back to spherical coordinates as follows

\( \rho = \sqrt {x^2 + y^2 + z^2} \) , \( \tan \theta = \dfrac{y}{x} \) , \( \cos \phi = \dfrac{z}{\sqrt {x^2 + y^2 + z^2}} \)

with \( 0 \le \theta \lt 2\pi \) and \( 0 \le \phi \le \pi \)