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 √R2−2RrLet O be the circumcenter and OF be the perpendicular to ABLet I be the incenter and IE be the perpendicular to AC Then,∠OAF=90∘−C ⇒∠OAI=∠IAF−∠OAF =A2−(90∘−C)=A2+C−A+B+C2=C−B2 Also,AI=IEsinA2=rsinA2=4RsinB2sinC2Hence in ΔOAI,OI2=OA2+AI2−2OAAIcos∠OAI =R2+16R2sin2B2sin2C2−8R2sinB2sinC2cosC−B2 ⇒OI2R2=1+16sin2B2sin2C2−8sinB2sinC2(cosB2cosC2+sinB2sinC2) =1−8sinB2sinC2(cosB2cosC2−sinB2sinC2)=1−8sinB2sinC2cosB+C2 =1−8sinB2sinC2sinA2 ∴OI=R√1−8sinA2sinB2sinC2=√R2−2Rr

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