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Question

In ABC, show that acosA+bcosB+ccosCa+b+c=rR

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Solution

Using a=2RsinA, b=2RsinB, c=2RsinC
So, acosA+bcosB+ccosCa+b+c
=R(sin2A+sin2B+sin2Ca+b+c)
using sin2A+sin2B+sin2C=4sinAsinBsinC
so, we get =4R(sinA+sinB+sinCa+b+c)
=abc2R2(2S)
=4R4R2S
so we get RS=rR

acosA+bcosB+ccosCa+b+c=rR
Hence, proved.


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