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Question

If in a ABC, b+ca=cotA2, show that ABC is a right-angled.

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Solution

In ABC,
b+ca=cotA2
2RsinB+2RsinC2RsinA=cosA2sinA2
sinB+sinC2sinA2cosA2=cosA2sinA2
2sinB+C2cosBC22cos180BC2=sin(90A2)
sinB+C2cosBC2sinB+C2=sinB+C2
sin(90BC2)=sinB+C2
180B+C=B+C2B=180B=90
So ABC is a right angled triangle

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