三角形ABC中,BC=a,AC=b,AB=c,则使等式{sin(A/2)}平方+{sin(B/2)}平方+{sin(C/2)}平方={cos(B/2)}平方,成立的充要条件是---答案是c+a=2b

Question

三角形ABC中,BC=a,AC=b,AB=c,则使等式{sin(A/2)}平方+{sin(B/2)}平方+{sin(C/2)}平方={cos(B/2)}平方,成立的充要条件是---
答案是c+a=2b

Answer 1

{sin(A/2)}²+{sin(B/2)}²+{sin(C/2)}²={cos(B/2)}²
<=>(1-cosA)/2+(1-cosB)/2+(1-cosC)/2=(1+cosB)/2
<=>2=cosA+2cosB+cosC
<=>2-2cosB=cosA+cosC
<=>4sin²(B/2)=2cos(A+C)/2cos(A-C)/2(因A+C=π-B,sinB/2>0)
<=>4sin²(B/2)=2sin(B/2)/2cos(A-C)/2(sinB/2>0)
<=>2sin(B/2)=cos(A-C)/2(cos(B/2)>0)
<=>4sin(B/2)cos(B/2)=2cos(B/2)cos(A-C)/2=2sin(A+C)/2cos(A-C)/2
<=>2sinB=sinA+sinC
<=>2b=a+c
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