Prove that there exists $6$ pairs of tangent circles
Source: Junior Olympiad of Malaysia Shortlist 2015 G7
July 17, 2015
geometry
Problem Statement
Let ABC be an acute triangle. Let HA,HB,HC be points on BC,AC,AB respectively such that AHA⊥BC,BHB⊥AC,CHC⊥AB. Let the circumcircles AHBHC,BHAHC,CHAHB be ωA,ωB,ωC with circumcenters OA,OB,OC respectively and define OAB∩ωB=PAB=B. Define PAC,PBA,PBC,PCA,PCB similarly. Define circles ωAB,ωAC to be OAPABHC,OAPACHB respectively. Define circles ωBA,ωBC,ωCA,ωCB similarly.Prove that there are 6 pairs of tangent circles in the 6 circles of the form ωxy.