2013年12月7日土曜日



△ABC において，BC の中点をＭとすると
ＡＢ^2＋ＡＣ^2＝ 2(ＡＭ^2＋ＢＭ^2）
が成り立ち、中線定理（パップスの定理）といいます。

Proof[edit]

Proof of Apollonius' theorem
The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (see parallelogram law). The following is an independent proof using the law of cosines.[1]
Let the triangle have sides a, b, c with a median d drawn to side a. Let m be the length of the segments of a formed by the median, so m is half of a. Let the angles formed between a and d be θ and θ′ where θ includes b and θ′ includes c. Then θ′ is the supplement of θ and cos θ′ = −cos θ. The law of cosines for θ and θ′ states
\begin{align} b^2 &= m^2 + d^2 - 2dm\cos\theta \\ c^2 &= m^2 + d^2 - 2dm\cos\theta' \\ &= m^2 + d^2 + 2dm\cos\theta.\, \end{align}
Add these equations to obtain
b^2 + c^2 = 2m^2 + 2d^2\,

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