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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. The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: [1]
A right triangle with the hypotenuse c. In a right triangle, the hypotenuse is the side that is opposite the right angle, while the other two sides are called the catheti or legs. [7] The length of the hypotenuse can be calculated using the square root function implied by the Pythagorean theorem.
A special trapezoid is an isosceles trapezoid with three equal sides, each longer than the fourth side, inscribed in the curve with a vertex ordering consistent with the clockwise ordering of the curve itself. Its size is the length of the part of the curve that extends around the three equal sides.
The celebrated Pythagorean theorem (book I, proposition 47) states that in any right triangle, 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 whose sides are the two legs (the two sides that meet at a right angle).
In a right triangle with legs a and b and altitude h from the hypotenuse to the right angle, h 2 is half the harmonic mean of a 2 and b 2. [17] [18] Let t and s (t > s) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then s 2 equals half the harmonic mean of c 2 and t 2.
Liu called the figure of the drawn diagram for the theorem the "diagram giving the relations between the hypotenuse and the sum and difference of the other two sides whereby one can find the unknown from the known." [6] In the field of plane areas and solid figures, Liu Hui was one of the greatest contributors to empirical solid geometry.
Possible geometric basis for a solution of IM 67118. Solid lines of the figure show stage 1; dashed lines and shading show stage 2. The central square has side b − a. The light gray region is the gnomon of area A = ab. The dark gray square (of side (b − a)/2) completes the gnomon to a square of side (b + a)/2.
They are also exactly the prime numbers such that there exists a right triangle with integer sides whose hypotenuse has length . For, if the triangle with legs x {\displaystyle x} and y {\displaystyle y} has hypotenuse length p {\displaystyle {\sqrt {p}}} (with x > y {\displaystyle x>y} ), then the triangle with legs x 2 − y 2 {\displaystyle ...