looking at the archives, I've looked at a discussion about the
notation for the cevian quotient.
Here is an (original?) argument for the notation P/M :
Consider an isocubic with pivot P, note multiplicatively the group
law of the cubic and take P as 1, then
For any point M on the cubic, P/M is the inverse of M in the group -
hence, it seems to be a quite correct notation -.
Let M*(pyz, qzx, rxy ) be the isoconjugate of M(x,y,z); P =(u,v,w)
P/M = [x (-x/u + y/v + z/w), ...].
Then det(M,P/M,P*) = -2 det(M, M*, P)
Thus M lies on the cubic <=> the line M,P/M goes through P*
<=> P/M lies on the cubic (with P/M instead of M)
Hence, if M lies on the cubic, P/M lies on the cubic and the line
M,P/M intersects again the cubic at P*; as the line PP* touches the
cubic at P, the proof is complete.
I think that it is often very interesting to consider an isocubic
with pivot P as the locus of M such as the line M,P/M goes through a
fixed point (P*)