Euclid's Pythagorean Theorem

Remember in like grade 8 when you learned about the Pythagorean Theorem? a squared plus b squared equals c squared for any right angle triangle. Simple enough, but if your class was like my class you didn't learn why this theorem is true. Well today Via De Scientia proves the Pythagorean Theorem. The ancient philosopher Euclid lived in Alexandria around 300 years before the Common Era. Euclid wrote a series of books called Elements in which he proves a variety of geometrical theorems. Euclid's work was very different from the math we know today because in Euclid's time algebra hadn't been invented yet. No one ever wrote an equation, graphed a function, or used a variable. All Euclid had was a straight edge and a compass. So all he could do was draw straight lines and perfect circles. Using those two techniques Euclid was able to design methods for constructing various objects and to prove a number of theorems that are still extremely relevant today. Euclid's version of the Pythagorean Theorem is devoid of equations because algebra hadn't been invented yet. Rather it states in right angled triangles the square on the side subtending the right angle is equal to the sum of the squares on the side containing the right angle. In plain English Euclid says that if we construct a square on each side of a right triangle then the two smaller squares always add up to the area of the bigger square. In this case the area of the red square plus the area of the blue square equals the area of the pink square. Now let's prove it. First we will give the points on the triangle names. The points are labeled A, B, and C such that the angle BAC is a right angle. So the theorem claims that the square of AB plus the square of AC equals the square of BC. We can construct the pink square BE which is just the square of the hypotenuse. Euclid had a particular method to do this using only a straight edge and compass but for the sake of keeping our diagram simple we will just assume this is possible. We can use the same method to construct the other two squares AF and AK here shown in red and blue respectively. Now we have our setup and we can start the proof. We now draw a line from A that is perpendicular to the hypotenuse perpendicular to BC. This line must intersect the segment DE at some point so let's give that point a name and call it L. We will now use a straight line segment to connect points A and D to form line segment AD. We can also connect C and F the same way. Notice that CBD and ABF are both right angles so if we add angle ABC to each of them they would still be equal. We thus know that angle DBA is equal to the angle CBF. Because the pink and red squares are well squares we know that BD is equal to BC and that AB is equal to BF. If two triangles have the same two sides at the same angle they must be equal triangles so we know the triangle ABD is equal to triangle BCF. Now consider this the area of a parallelogram is base times height and the area of a triangle is base times height over 2. That means that if a parallelogram and a triangle have the same base and height the triangle has half the area of the parallelogram no matter what the angles are. That means that the area of the triangle ABD is exactly half the area of the parallelogram BL which shares the base BD. That also means that the area of the triangle BCF is half the area of the parallelogram BG which shares the base BF. So that means because the two triangles are equal that the area of the rectangle BL is the same as the area of the square BG. We can then use the same method by drawing these lines to show that this triangle is equal to this triangle. Similarly this allows us to show that the areas of these two parallelograms are equal. So we've shown that the red square has the same area as the red rectangle and the blue square has the same area as the blue rectangle. But we also know that the red rectangle and the blue rectangle add up to the pink square. Thus we have shown that the areas of the red square and the blue square add up to the area of the pink square. A squared plus B squared equals C squared which is exactly what we were trying to show. Thanks for watching! I hope you enjoyed learning why the Pythagorean theorem is as right as the triangles it describes. As you can see equations are not necessary to do good math. Geometry is more than equal to the task. Do you have any feedback on this video or suggestions for future videos? Then leave them in the comment section below. Also don't forget to press the like button and the subscribe button!