Let P be a point, 1,2,3,4 four lines passing through P and 0 a line intersecting the four lines at four distinct and real points (ie not passing through P and not parallel to some one of the four lines)
r_ij := the inradius of the triangle bounded by the lines (0,i,j)
[(1/(r_12*r_24)) - (1/(r_13*r_34))] /
[((1/r_12) + (1/r_24)) - ((1/r_13) + (1/r_34))]
Simple application of the altitude formula found HERE.
Exercise for the reader:
Find the formula of the r_23