Re: [中學] 100台南二中教甄證明
※ 引述《tingisall ( @@)》之銘言：
: 三角形ABC中 AB=c BC=a AC=b
: 已知tan(A/2)‧tan(C/2)=1/3
: 證:a b c三數成等差
: (題目不知道有沒有記錯 有錯請指正 謝謝)
設 x=(-a+b+c)/2, y=(a-b+c)/2, z=(a+b-c)/2
條件即為(r/x)(r/z)=1/3
ABC面積=((x+y+z)xyz)^(1/2)=r(x+y+z)
=> (x+y+z)xyz=r^2(x+y+z)^2=(1/3)xz(x+y+z)^2
=> 3y=x+y+z => (y+z)+(x+y)=2(x+z) => a,b,c成等差
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