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The '''Pythagorean theorem''' is possibly the most well known mathematical theorem. It states that squaring the lengths of the two smaller sides of a [[right triangle]] and adding them together will get the length of the [[hypotenuse]] squared. This is expressed mathematically as:
The '''Pythagorean theorem''' is possibly the most well known mathematical theorem. It states that squaring the lengths of the two smaller sides of a [[right triangle]] and adding them together will get the length of the [[hypotenuse]] squared. This is expressed mathematically as:
<math>a^2+b^2=c^2</math>
<math>a^2+b^2=c^2</math> or, <math>c=\sqrt{a^2+b^2}</math>
The Pythagorean Theorem was developed by the [[Greece|Greek]] mathematician [[Pythagoras]]. Is is a mathematical relationship between the lengths of the sides of a right triangle which can be illustrated as follows:
This identity only holds true for a right triangle drawn on a flat ([[Euclid]]ean) plane
The Pythagorean Theorem was developed by the [[Greece|Greek]] mathematician [[Pythagoras]]. The theorem describes a mathematical relationship between the lengths of the sides of a right triangle which can be illustrated as follows:
[[Image:Pythagorean_Theorem.png|left|350px|Pythagorean Theorem]]
[[Image:Pythagorean_Theorem.png|left|350px|Pythagorean Theorem]]
<br>
<br>
<center>[[image:Pythagoras.gif]]</center>
<center>[[image:Pythagoras.gif]]</center>
__NOTOC__
==Corollary: No three lines with odd prime numbers as lengths can form a right triangle==
Indirect proof: Let's describe a situation where lines with the lengths of three odd prime numbers ''do'' form a right triangle and see what follows.
===Conceptualization===
Let's call the two odd prime legs ''l''<sub>1</sub> and ''l''<sub>2</sub> and ''h'' the odd prime hypoteneuse.
One definition of odd prime number ''op'' is that it is an odd natural number with no positive integral factor ''f'' other than one and itself.
===Hypothesis===
:Simple assumption: The hypoteneuse is always larger than either of its lengths.
That is, it is to be proven that the first proposition (that is, the proposition that is the simple assumption, the ''assumption for a conditional proof'') leads to the second.
===Conceptualization (cont'd)===
The conceptualization of the Pythagorean theorem in the case of the three odd prime numbers, that is using the ''assumption for an indirect proof'' is therefore: ''l''<sub>1</sub><sup>2</sup> + ''l''<sub>2</sub><sup>2</sup> = ''h''<sup>2</sup>.
====Reasoning====
1. Let's divide one of the two odd prime number "''h''" factors contained in the "''h''<sup>2</sup>" by both sides of the equation:
:(''l''<sub>1</sub><sup>2</sup> / ''h'') + (''l''<sub>2</sub><sup>2</sup> / ''h'') = ''h''.
2. And since ''l''<sub>1</sub> and ''l''<sub>2</sub> are whole numbers, that which results when they are multiplied by themselves, namely ''l''<sub>1</sub><sup>2</sup> and ''l''<sub>2</sub><sup>2</sup>, certainly are as well.
[[Category:Geometry]]
[[Category:Plane Geometry]]
[[Category:Mathematics]]
[[Category:Mathematics]]
[[Category:Geometric Theorems]]