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Question

In Δ ABC the sides opposite to angles A,B,C are denoted by a,b,c respectively. Let r and R be the inradius and the circumradius of the triangle, then the distance between the circumcenter and the incenter of ΔABC is

A
R22Rr
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B
R2Rr
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C
R22R
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D
R22r
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Solution

The correct option is A R22Rr
Let O be the circumcenter and OF be the perpendicular to AB
Let I be the incenter and IE be the perpendicular to AC
Then,OAF=90C OAI=IAFOAF =A2(90C)=A2+CA+B+C2=CB2
Also,AI=IEsinA2=rsinA2=4RsinB2sinC2
Hence in ΔOAI,OI2=OA2+AI22OAAIcosOAI =R2+16R2sin2B2sin2C28R2sinB2sinC2cosCB2 OI2R2=1+16sin2B2sin2C28sinB2sinC2(cosB2cosC2+sinB2sinC2) =18sinB2sinC2(cosB2cosC2sinB2sinC2)
=18sinB2sinC2cosB+C2 =18sinB2sinC2sinA2 OI=R18sinA2sinB2sinC2=R22Rr

366604_148283_ans_a1996a2544154cb3bdd7f796f0455940.png

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