Concyclic Points from Reflections of the Circumcenter about the Altitudes
Initializing live version

Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
Let ABC be a triangle with circumcenter O. Let A'', B'', and C'' be the reflections of O in the altitudes AA', BB', and CC', respectively. Then A'', B'', C'', and O are concyclic.
Contributed by: Jay Warendorff (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
See figure 3 in Two Triads of Congruent Circles from Reflections.