MATH 392 -- Geometry Through History 

 

Euclid's Proof of the Pythagorean Theorem (Proposition I.47 of the Elements) 

 

The following animations give the idea behind Euclid's ingenious  

and elegant proof of the Pythagorean Theorem from Proposition 47 in  

Book I of the Elements.  Note:  Many historians of mathematics   

believe that this proof was probably an original contribution of Euclid  

since this proof does not appear elsewhere in ancient Greek mathematical   

texts.
 

The red figures are the squares on the sides of a given right triangle  

(a 3-4-5  right triangle with hypotenuse along the  x-axis). 

The dashed blue line shows a perpendicular dropped from the vertex  

opposite the hypotenuse to the hypotenuse.  Extended, this line  

subdivides the square on the hypotenuse into two rectangles.  The  

idea of the proof is to show that the square on the other sides of  

the right triangle have areas equal to these two rectangles.  Establishing  

this will show that the square on the hypotenuse is equal to the   

sum of the squares on the other two sides,  which is Euclid's form  

of the theorem.  

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Maple Commands  

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Plot_2d
 

 

In the first animation (above), consider the black triangle that is  

formed by a diagonal of the square to the left of the blue line.  

We keep the base along a side of the small square, but slide  

the upper vertex along the parallel line.  From Proposition I.37, we   

know that all these triangles have equal area.  We continue sliding  

the upper vertex until it reaches the far end of the hypotenuse. 

(The animation is "looped," though so the whole process repeats.) 

 

 

The next animation rotates the final position of the black triangle by  

90 degrees clockwise about the point B, demonstrating visually that 

triangles FBC and ABD are congruent: 

 

> display(Pyth, Sub, Anim3, Labels, view = [-3 .. 8, -6 .. 6]); 1
 

Plot_2d
 

 

Using Proposition I.37 again, we slide one vertex along the dashed blue  

line until it reaches the bottom of the rectangle on the left.  All of these  

triangles also have the same area.   

 

> display(Pyth, Sub, Anim2, Labels, view = [-3 .. 8, -6 .. 6])
 

Plot_2d
 

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The final triangle is half of the left rectangle.  Hence half of the   

small square equals half that rectangle.  So the whole square also  

equals the whole rectangle. 

 

Finally, we do a similar argument with the other square and the rectangle  

on the right of the blue line (not shown, but the idea is the same!) to finish  

the proof.  Pretty nifty!