Leonardo da Vinci pythagorean theorem proof

Pythagorean Theorem Proof

David Blake, Dallon Yeager, Dallin Fuell, Jeff Arens

Leonardo Da Vinci's proof.

Leonardo Was born April 15 of 1452

in Florence, Italy

Not only was he a mathematician, he was an artist, a scientist, and a philosopher.

His full name is Leonardo di ser Piero da Vinci

Leonardo da Vinci did many things in his lifetime, including contributing to math and science and painting many works of art such as the Mona Lisa.

People Called him a man of "unqenchable curiosty" and

He is widely considered to be one of the worlds most diversely talented person who has ever lived.

Leonardo da Vinci came up with the concept of several modern day inventions such as flying machines, waepons, and instruments.

Leonardo Da Vinci also painted many famous paintings such as, the mona lisa, the last supper, and St john in the wilderness.

The Mona Lisa

Leonardo passed away at Clos Luce, May 2, 1519.

Leonardo Da Vinci influenced many of our modern technology with his drawings of his ideas.

Although he is widely known for his paintings, he did create this pythagorean theorem proof.

"This proof is ascribed to Leonardo da Vinci (1452-1519) [Eves]. Quadrilaterals ABHI, JHBC, ADGC, and EDGF are all equal. (This follows from the observation that the angle ABH is 45°. This is so because ABC is right-angled, thus center O of the square ACJI lies on the circle circumscribing triangle ABC. Obviously, angle ABO is 45°.) Now, Area(ABHI) + Area(JHBC) = Area(ADGC) + Area(EDGF). Each sum contains two areas of triangles equal to ABC (IJH or BEF) removing which one obtains the Pythagorean Theorem."

-http://www.cut-the-knot.org/pythagoras/

In this figure triangle BAD + triangle BGC is congruent to quadrilateral CJKL

In order to use the figure below to prove the theorem a² + b² = c², we need to first identify the length of sides using the a b and c. In using triangle 1, we can label the short side a, the long side b, and the hypotenuse c. After these sides are established, we can then use the line of symmetry to identify and compare the lengths of the other sides, labeling them as either a b or c as they correspond.

After identifying all sides as either a b or c, we then have to turn our focus to the area. For this, it is necessary to identify sections of similar areas using the dotted lines. These sections are labeled 1, 2, 3, and 4.

Since all of these sections contain the same side lengths of a b and c we can conclude that the length of the two dotted lines is equal. Thus, the areas of the sections are equal as well. Using these 4 sections, we can combine them into two equal area sections, shown in figures 1 and 2.

Notice that both sections 1 and 2 share the shaded region.

If we pull the figures apart so that they do not overlap, we can see that figure 1 and figure 2 both have two triangles of equal area: triangles abc. Due to this, we can remove them both from each section. This leaves us with two different groups of figures with the same area, since the same amount of area was removed from each

Thus, we can see that the two squares on the left have a total area of a² + b². Likewise, we can see that the square on the right has an area of C².

Therefore: a²+b²=c²

Sources:

http://en.wikiversity.org/wiki/Pythagorean_Theorem_Proof_by_Picture-Leonardo_da_Vinci_Proof

-http://www.cut-the-knot.org/pythagoras/

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Leonardo da Vinci's Proof

http://mail.mhrd.k12.nj.us/~lsantucci/FOV1-0001258E/FOV1-00012591/FOV1-00013DE0/Leonardo%20Proof%20exploration.pdf?FCItemID=S002E1A57