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13.4

Part of Geometry Mathley 2011-12

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Geometry Mathley 13.4 concurrent circles, similar triangles

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6/7/2020
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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similarconcurrent circlesconcurrentcirclesgeometrysimilar triangles