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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle ) is equal to the sum of the areas of the squares on the other two sides.
勾股定理（英語： Pythagorean theorem / Pythagoras' theorem ）是平面几何中一个基本而重要的定理。 勾股定理说明， 平面 上的 直角三角形 的两条直角边的长度（较短直角边古称勾长、较长直角边古称股长）的 平方 和等于斜边长（古称弦长）的平方。
- 定理
- 其他形式
- 畢氏三元數組
- 證明
- 畢氏定理的逆定理
- 外部連結
在平面上的一個直角三角形中，兩個直角邊邊長的平方加起來等於斜邊長的平方。如果設直角三角形的兩條直角邊長度分別是a {\displaystyle a} 和b {\displaystyle b} ，斜邊長度是c {\displaystyle c} ，那麼可以用數學語言表達： 1. 1.1. a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} 或 1. 1.1. a 2 + b 2 = c {\displaystyle {\sqrt {a^{2}+b^{2}}}=c} 餘弦定理是畢氏定理的一個推廣。畢氏定理現約有400種證明方法，是數學定理中證明方法最多的定理之一。
如果c {\displaystyle c} 是斜邊的長度而a {\displaystyle a} 和b {\displaystyle b} 是另外兩條邊的長度，畢氏定理可以寫成： a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}\,} 如果a {\displaystyle a} 和b {\displaystyle b} 知道，c {\displaystyle c} 可以這樣寫： c = a 2 + b 2 . {\displaystyle c={\sqrt {a^{2}+b^{2}}}.\,} 如果斜邊的長度c {\displaystyle c} 和其中一條邊（a {\displaystyle a} 或b {\displaystyle b} ）...
畢氏三元數組是滿足畢氏定理a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} 的正整數組( a , b , c ) {\displaystyle (a,b,c)} ，其中的a , b , c {\displaystyle a,b,c} 稱為畢氏三元數。例如( 3 , 4 , 5 ) {\displaystyle (3,4,5)} 就是一組畢氏三元數組。 任意一組畢氏三元數( a , b , c ) {\displaystyle (a,b,c)} 可以表示為如下形式：a = k ( m 2 − n 2 ) , b = 2 k m n , c = k ( m 2 + n 2 ) {\displaystyle a=k(m^{2}-n^{2}),b=2...
這個定理有許多證明的方法，其證明的方法可能是數學眾多定理中最多的。路明思（Elisha Scott Loomis）的Pythagorean Proposition一書中總共提到367種證明方式。 有人會嘗試以三角恆等式（例如：正弦和餘弦函數的泰勒級數）來證明畢氏定理，但是，因為所有的基本三角恆等式都是建基於畢氏定理，所以不能作為畢氏定理的證明（參見循環論證）。
畢氏定理的逆定理是判斷三角形為鈍角、銳角或直角的一個簡單的方法，其中A B ¯ = c {\displaystyle {\overline {AB}}=c} 為最長邊: 1. 如果a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}\,} ，則△ A B C {\displaystyle \triangle ABC} 是直角三角形。其中∠ C {\displaystyle \angle C} 是直角。 2. 如果a 2 + b 2 > c 2 {\displaystyle a^{2}+b^{2}>c^{2}\,} ，則△ A B C {\displaystyle \triangle ABC} 是銳角三角形（若無先前條件A B ¯ = c {\dis...
畢氏定理（MathWorld） （頁面存檔備份，存於網際網路檔案館）（英文）When a triangle has a right angle (90°) ... ... and squares are made on each of the three sides, ... ... then the biggest square has the exact same area as the other two squares put together! It is called "Pythagoras' Theorem" and can be written in one short equation: a 2 + b 2 = c 2. Note:
Pythagoras theorem explains the relation between base, perpendicular and hypotenuse of a right-angled triangle. Learn how to proof the theorem and solve questions based on the formula. Login
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The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be expressed as, c 2 = a 2 + b 2; where 'c' is the hypotenuse and 'a' and 'b' are the two legs of the triangle.
The Pythagorean theorem states that if a triangle has one right angle, then the square of the longest side, called the hypotenuse, is equal to the sum of the squares of the lengths of the two shorter sides, called the legs.
