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Three circles having common radical axis

Source: KöMaL A. 736

December 20, 2018
power of a pointradical axis

Problem Statement

Let PP be a point in the plane of triangle ABCABC. Denote the reflections of A,B,CA,B,C over PP by A′,B′A',B' and C′C', respectively. Let A′′,B′′,C′′A'',B'',C'' be the reflection of A′,B′,C′A',B',C' over BC,CABC,CA and ABAB, respectively. Let the line A′′B′′A''B'' intersects ACAC at AbA_b and let A′′C′′A''C'' intersects ABAB at AcA_c. Denote by ωA\omega_A the circle through the points A,Ab,AcA,A_b,A_c. The circles ωB,ωC\omega_B,\omega_C are defined similarly. Prove that ωA,ωB,ωC\omega_A ,\omega_B ,\omega_C are coaxial, i.e., they share a common radical axis.
Proposed by Navneel Singhal, Delhi and K. V. Sudharshan, Chennai, India
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