Problems says:
Given $2$ points $A,B$ find the bisector of segment $AB$.
(I hope I'm using correct terms, this was translated by myself. In the original text h-points,h-segment,h-bisector were used but I didn't find those in English literature)
Suppose one of the points is at the center of disc WLOG $A$
Text claims that line s which is a part of a circle that maps $B$ to $A$ through inversion is the bisector. I do understand that property of s but I don't understand how do we see that s bisects $AB$
Next picture shows how s in constructed
Can you explain why is s the bisector ?