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Math Help - Circumradius and inradius.

  1. #1
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    Circumradius and inradius.

    prove that for any triangle ,R>= 2r,where R and r are the circumradius and inradius of the triangle respectively.
    post in different proofs
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Not standard way...

    Let ABC be some triangle,with AB=a, AC=b, BC=c,and with area S.

    Let, p=(a+b+c)/3, hence, r=S/p.

    We, also know that: R=abc/4S

    I'ii leave like that, maybe someone would like to proceed somehow(if possible)...
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  3. #3
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    THANKS!
    oh yes!i wish to complete your proof:
    Let ABC be some triangle,with AB=a, AC=b, BC=c,and with area S.

    Let, p=(a+b+c)/2, hence, r=S/p.

    We, also know that: R=abc/4S
    now we just have to show that abc/4s >= 2s/p ,which reduces to abc>= 8(p-a)(p-b)(p-c). now we know that a^2-(b-c)^2<a^2.so similarly we could write
    abc>=sqrt( a^2-(b-c)^2)sqrt(b^2-(c-a)^2)sqrt(c^2-(a-b)^2)
    =(a+b-c)(b+c-a)(c+a-b)=8(p-a)(p-b)(p-c) Q.E.D
    Not standard way...
    please show the standard way,it will be very helpful
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  4. #4
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    hey!
    post in other proofs.is there a proof with the help of the eulers theorem: (SI)^2=R^2-2Rr,Where S,I,R,R are the circumcenter,incenter,circumradius,inradius respectively
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