截線定理

截線定理(英語:Intercept theorem),是平面幾何中的基本定理之一。截線定理說明,平面上的一個三角形中,若在其中一條腰的中點作一條直線,與其底邊平行,則該線穿過另一條腰的中點。這定理可推廣到梯形上,以及一般化至任意分割比例的情況。截線定理與另外兩條幾何定理中點定理等比定理有密切關係。

定理

截線定理的最基本形式是在三角形上的應用。

圖中有三角形 ,作一條直線 與底邊 平行。

截線定理說明,若 ,則

換句話說, 是三角形 中位線

這定理能簡單推廣到梯形上應用。

圖中有梯形 ,其中 。作一條直線 與上底 和下底 平行。

截線定理說明,若 ,則

同樣地, 是梯形 的中位線。

一般化定理

對於平行線將腰分割成任意比例的情形,一般化截線定理則給出,左右兩條腰的分割比例相等。

在上圖的三角形 中,若 ,則有

同樣地,在梯形 ,若 ,則有

證明

這定理能以相似三角形簡單證明。


考慮上圖的 。由於

  • (公共角)
  • (平行線的同位角)
  • (平行線的同位角)

所以。(等角)

由此可得 。(相似三角形的對應邊)

因此

證畢。


對於梯形的情況,考慮梯形 ,在 上作一直線,與 平行,並與 分別相交於

由定義可知,平行四邊形

因此 。(平行四邊形的對邊)

上面已證明,由 ,可知

代入可得

證畢。

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