Three Congruent Circles by Reflection III: What is this about?
A Mathematical Droodle
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Copyright © 19962018 Alexander BogomolnyThe applet attempts to illustrate a theorem by Quang Tuan Bui
Given ΔABC and a point P with the pedal ΔP_{a}P_{b}P_{c}, as usual. Let B_{a} and C_{a} be reflections of B in P_{c} and C in P_{b}, respectively. (O_{a}) is the circumcircle of ΔAB_{a}C_{a}. The points A_{b}, C_{b}, A_{c}, B_{c}, and the circles (O_{b}) and (O_{c}) are defined similarly. Then the three circles (O_{a}), (O_{b}) and (O_{c}) are congruent. 
What if applet does not run? 
Note that the theorem generalizes two previous results where the vertices of a triangle have been reflected in the altitudes and its angle bisectors. The former is obvious. The latter stems from an observation that, if I denotes the incenter of ΔABC with pedal points I_{a}, I_{b} and I_{c}, the reflection in IA of the reflection in IC of B is the reflection of B in II_{c}.
Proof of the theorem
We proceed very much in the spirit of the other two statements. Let O and O(P) be the circumcenter of ΔABC and the circle through P with center O. Introduce further

Naturally, X, Y, and Z all lie on O(P), so that, for example, PP_{a}A_{p}X is a rectangle with one midline through O. We may also conclude that (O_{a}) is the reflection of O(P) in A' and similarly for (O_{b}) and (O_{c}) from which it follows that the latter three are indeed congruent and have the radius of OP.
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Copyright © 19962018 Alexander Bogomolny72023521