加菲證明畢氏定理

我又做標題黨了,這個加菲並不是加菲貓,而是美國第二十任總統占士.加菲 (James Abram Garfield)。

畢氏定理說,一個直角三角形中,直角 (即 90 度) 的兩條鄰邊長度各自二次方相加等於斜邊長度的二次方:

Screen Shot 2015-06-20 at 17.33.05

[上圖應為「畢氏」,懶改,注意。謝 Alan Chiu 指正!]

加菲總統的證明很簡單,我們只需要知道幾件事實:

  1. 三角形內角總和是 180 度;
  2. 直線上的角度總和是 180 度;
  3. 三角形、正方形、梯形面積計算方法。

以上事實的證明十分簡單,就留給有興趣的讀者自行證明吧!事不宣遲,我們來證明畢氏定理吧:

我們首先沿 a 邊向上畫多一個一樣的直角三角形,不過這次把它順時針轉 90 度來畫。最後,我們好像下圖中把右上角和右下角的兩點連起來,得到了一個梯形:

Screen Shot 2015-06-20 at 17.28.00

現在,我們問,圖中叫做 x 的角度是多少?給大家一分鐘。

答案是 90 度,即是一個直角。為什麼?如果叫 c 邊與 b 邊的夾角做 y,那麼 c 邊與 a 邊的夾角就是 90 度 – y,因為三角形內角總和是 180 度。接著,看看左邊垂直的 (a + b) 邊,因為直線上的角度總和亦是 180 度,我們就有

Screen Shot 2015-06-20 at 17.28.35

Screen Shot 2015-06-20 at 17.28.14

所以,圖中打斜的是一個等腰直角三角形,兩條邊長是 c。換句話說,這個三角形的面積就是一個邊長為 c 的正方的一半。

我們就知道這個梯形的面積是兩個細直角三角形的面積再加上邊長是 c 的等腰直角三角形的面積。而且,我們也知道梯形的面積是「上底加下底乘高除二」。所以

Screen Shot 2015-06-20 at 17.28.19

畢氏定理得證。

封面圖片來源:http://www.larouchepub.com/eiw/public/1995/eirv22n09-19950224/eirv22n09-19950224_050-garfield_the_pythagorean_theorem.pdf

By David Yu

David Yu is the father of Simba and an Astrophysicist. He obtained his PhD in Astrophysics from the Max Planck Institute of Extraterrestrial Physics, Germany. He was formerly a postdoc at the KTH Royal Institute of Technology, Sweden, and is currently a lecturer at the University of Hong Kong. He served as the host of the RTHK TV31 programme Science Night, the Depute Editor-in-Chief of the Taiwanese Science Magazine Physics Bimonthly, and the Science Consultant of Stand News. The books he authored include Dr David’s 21 Mysteries of The Universe (2021), Space – A Down-to-Earth Astrophysicist (2021), Time – A Spacetime Journey of Astronomy and Physics (2019), and Secrets from the Stars (2017).

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