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平行線分三角形梯形的對角線交點在三角形中線

2023-02-22 06:25 作者:獅子朗道 0人讀過 | 我要投稿

也即，三角形作底邊平行線與另外兩邊構(gòu)成梯形。這個梯形的對角線交點在三角形中線上。

證明：

假設(shè)G不為BC中點， $%5Cfrac%7BBG%7D%7BGC%7D%3D%5Cfrac%7B1%7D%7Bk%7D%5Cneq%7B1%7D$ .

我們有等高三角形的面積比 $S_%7B%5Cbigtriangleup%20ABG%7D%3AS_%7B%5Cbigtriangleup%20ACG%7D%3D1%3Ak%2C%5Cquad%20S_%7B%5Cbigtriangleup%20FBG%7D%3AS_%7B%5Cbigtriangleup%20FCG%7D%3D1%3Ak$ .

而 $%5Cbigtriangleup%7BABG%7D%5Cbacksim%5Cbigtriangleup%7BADH%7D$ ,? $%5Cbigtriangleup%7BACG%7D%5Cbacksim%5Cbigtriangleup%7BAEH%7D.$

我們有

$DH%3D%5Cfrac%7BAH%7D%7BAG%7DBG%2C%5Cquad%7B%7DEH%3D%5Cfrac%7BAH%7D%7BAG%7DCG%5Cquad%7B%7D%2C%0A$

$DH%3AEH%3D1%3Ak.$

故而類似的等高三角形面積比，

$S_%7B%5Cbigtriangleup%20ADH%7D%3AS_%7B%5Cbigtriangleup%20AEH%7D%3D1%3Ak%2C%5Cquad%20S_%7B%5Cbigtriangleup%20FDH%7D%3AS_%7B%5Cbigtriangleup%20FEH%7D%3D1%3Ak.%0A$

故而

$%0A%5Cfrac%7BS_%7B%5Cbigtriangleup%20DBF%7D%7D%7BS_%7B%5Cbigtriangleup%20ECF%7D%7D%3D%5Cfrac%7BS_%7B%5Cbigtriangleup%20ABG%7D-S_%7B%5Cbigtriangleup%20ADH%7D-S_%7B%5Cbigtriangleup%20FDH%7D-S_%7B%5Cbigtriangleup%20FBG%7D%7D%7BS_%7B%5Cbigtriangleup%20ACG%7D-S_%7B%5Cbigtriangleup%20AEH%7D-S_%7B%5Cbigtriangleup%20FEH%7D-S_%7B%5Cbigtriangleup%20FCG%7D%7D%3D%5Cfrac%7B1%7D%7Bk%7D%5Cneq%201%20%5Ctag%7B1%7D$

而我們知道，梯形對角線分左右兩個小三角形面積應(yīng)該是相等的. 詳細(xì)如下（證明）.

對等底等高的三角形有

$%5Cbegin%7Balign*%7D%0A%0AS_%7B%5Cbigtriangleup%20DBC%7D%26%3DS_%7B%5Cbigtriangleup%20ECB%7D%20%5C%5C%0A%0AS_%7B%5Cbigtriangleup%20DBC%7D-S_%7B%5Cbigtriangleup%20FBC%7D%26%3DS_%7B%5Cbigtriangleup%20ECB%7D-S_%7B%5Cbigtriangleup%20FCB%7D%20%5C%5C%0A%0A%7BS_%7B%5Cbigtriangleup%20DBF%7D%7D%26%3D%7BS_%7B%5Cbigtriangleup%20ECF%7D%7D%20%5Ctag%7B2%7D%0A%0A%5Cend%7Balign*%7D$