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Geometry Mathley 13.4 concurrent circles, similar triangles

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June 7, 2020
similarconcurrent circlesconcurrentcirclesgeometrysimilar triangles

Problem Statement

Let PP be an arbitrary point in the plane of triangle ABCABC. Lines PA,PB,PCPA, PB, PC meets the perpendicular bisectors of BC,CA,ABBC,CA,AB at Oa,Ob,OcO_a,O_b,O_c respectively. Let (Oa)(O_a) be the circle with center OaO_a passing through two points B,CB,C, two circles (Ob),(Oc)(O_b), (O_c) are defined in the same manner. Two circles (Ob),(Oc)(O_b), (O_c) meets at A1A_1, distinct from AA. Points B1,C1B_1,C_1 are defined in the same manner. Let QQ be an arbitrary point in the plane of ABCABC and QB,QCQB,QC meets (Oc)(O_c) and (Ob)(O_b) at A2,A3A_2,A_3 distinct from B,CB,C. Similarly, we have points B2,B3,C2,C3B_2,B_3,C_2,C_3. Let (Ka),(Kb),(Kc)(K_a), (K_b), (K_c) be the circumcircles of triangles A1A2A3,B1B2B3,C1C2C3A_1A_2A_3, B_1B_2B_3, C_1C_2C_3. Prove that (a) three circles (Ka),(Kb),(Kc)(K_a), (K_b), (K_c) have a common point. (b) two triangles KaKbKc,ABCK_aK_bK_c, ABC are similar.
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