Re: [中學] 矩形外接三角形的面積最小值

[mack (腦海裡依然記得妳)]

※ 引述《deryann (星辰)》之銘言：
: 三角形ABC內有一內接矩形PQRS..
: QR 在BC上
: P 在AB上
: S 在AC上
: 邊長為高度為PQ 3 寬度QR 為4
: 求此三角形面積最小值
: 謝謝各位!
做BC邊的高AD交PS於E點
假設 AD = x => AE = x-3
   BQ:PE = PQ:AE = 3:(x-3) = RS:AE = CR:SE
=> 3:(x-3) = (BQ+CR):(PE+SE) = (BQ+CR):4 => (BQ+CR) = 12/(x-3)
=> BC = BQ+CR+QR = 12/(x-3) + 4 = 4x/(x-3)

=>△ABC = x*4x/(x-3) ÷2 = 2x^2/(x-3) = (2x+6) + 18/(x-3)
                                      = 12 + 2(x-3) + 18/(x-3)
                                      ≧ 12 + 2√(2*18)
                                      = 12 + 12
                                      = 24

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