This appears to be the definition of conjugate points in projective geometry. Lets say we have a circle w and a ray, Oa, from the center of the circle, O, outward with a point on it, say a. To invert the point we find a new point, a', on this ray (Oa) such that (Oa)(Oa') = r^2. Likewise, starting from a' we map to a. The points inside the circle map to the points outside of the circle, points on the circle map to themselves, and infinity (ideal point)maps to the circles center. So, thats the basic for inversive geometry. However, for projective geometry a will map to the line, say A, passing through a' perpenicular to the ray. So, a point will map to a line and a line will map to a point. But its important to note that it passes through the inverse to the original point.
For that definition they have two rays from the circle w (O, r) with a and a' on one and b and b' on the other. They also have the lines A and B passing through a' and b' respectively. So, they are saying that the line A goes through b (which is on the other ray) iff B goes through the point a. This can be proven when you realise the perpendicular lines with the other rays form right triangles which are similar.
Does this make sense?