Significant points on circles centre the circumcentre

Extract

In Figure 1, triangle ABC has circumcentre O and X, Y, Z are the midpoints of the sides BC, CA, AB respectively. P is a point in the plane of ABC not lying on any of its sides. Σ is the circumcircle and S is the circle centre O passing through P.

Points on S are defined as follows. Points U, V, W are the points of intersection of AP, BP, CP with S. Points L, M, N are the reflections of P in the lines XO, YO, ZO respectively.