Re: [中學] 方程組一題
※ 引述《izenkuo (??????)》之銘言:
: x,y,z為正數,且
: x^2 + y^2 + xy = 25
: y^2 + z^2 + yz = 9
: z^2 + x^2 + zx = 16
: 求 x + y + z 之值為多少?
: 再麻煩各位高手了,感激不盡~~
作圖使得 PA = x, PB = y, PC = z,且角APB=角BPC=角CPA = 120度
則由餘弦定理:
AB = 5, BC = 3, CA = 4
此時P即ABC之費馬點,所求
x+y+z = PA + PB + PC = BX
其中X為在AC邊向外作正三角形的另一頂點
畫個圖,利用畢氏定理可得
所求 = sqrt( (3+ 2 sqrt(3))^2 + 2^2)= sqrt(25+12 sqrt(3))
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代數幾何觀點!
Algebro-Geometrical Aspect!
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