O is the circumcenter of ΔABC and R1,R2,R3 are respectively the radii of the circumcircles of the triangles OBA,OCA and OAB
Prove that aR1+bR2+cR3=abcR3
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Solution
If O is the circumcenter of ΔABC then OA=OB=OC=R Let R1,R2 and R3 be the circumradii of ΔOBCΔOCA and ΔOAB respectively. In ΔOBC2R1=asin2A or aR1=2sin2A Similarly, bR2=2sin2B and cR3=2sin2C ∴aR1+bR2+cR3=2(sin2A+sin2B+sin2C)=8sinAsinBsinC=8a2Rb2Rc2R=abcR3