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THE DAY THE MATH WORLD STOOD STILL. Or Euclid Saves Pythagoras. How did Euclid prove the Pythagorean Theorem. In Euclid’s time, Arabic numerals were unknown to the Greek world. Greeks knew geometry by shapes or units not numbers. So how did Euclid prove a 2 + b 2 = c 2 ?.
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THE DAY THE MATH WORLD STOOD STILL Or Euclid Saves Pythagoras
How did Euclid prove the Pythagorean Theorem • In Euclid’s time, Arabic numerals were unknown to the Greek world. • Greeks knew geometry by shapes or units not numbers. • So how did Euclid prove a2 + b2 = c2?
Euclid’s Drawing First, he began with a right triangle.
Euclid’s Drawing, Part II Next, he drew squares of the same size as the sides of the triangle.
E F A D G C B K I H J Euclid’s Next Steps • Then, he drew a perpendicular line from HI to A, denoted as J. • And another line from A to H and from C to G. • Then, the Area of rectangle BHJK = 2 area of triangle ABH, since their bases are equal and they live in the same parallels. • Likewise, the Area of square ABGF = 2x Area of triangle CBG.
E F A D G C B K I H J Euclid’s Steps .. Continued • Note that angle ABH = angle ABC + angle CBH and angle CBG = angle ABC + angle ABG. • Since angle ABG = angle CBH, then angle CBG = angle ABH. Also, BH = CB and AB = AB. • Then, area of rectangle BHJK = 2 area of triangle ABH = 2 area of triangle CBG = area of rectangle ABFG.
E F A D G C B I H Euclid’s Conclusion • Do the same to the other square. • Then, square ABFG + square ACDE = square CBHI. • This is Euclid’s proof of the Pythagorean Theorem. Q.E.D.
