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Garfield was the 20th President in 1881 and did this proof of the Pythagorean Theorem while he was still a seated member of Congress in 1876. It is interesting to note that he was fascinated by geometry, like President Lincoln, but was not a professional mathematician or geometer.
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Part 1 of 3:The Tutorial
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1Construct a right triangle resting on side b with right angle to the left connected to upright and perpendicular side a, with side c connecting the endpoints of a and b.,br>

2Construct a similar triangle with side b now extending in a straight line from the original side a, then with side a parallel along the top to the bottom original side b, and side c connecting the endpoints of the new a and b.

3Understand the goal. We are interested to know the angle x formed where the two side c's meet. Thinking about it, the original triangle was made of 180 degrees with the angle on the right at the far end of b, called theta, and the other angle at the top of a, being 90 degrees minus theta, as all the angles total 180 degrees and we already have one 90 degree angle.

4Transfer your angle knowledge to the upper new triangle. At the bottom, we have theta, at the top left we have 90 degrees, and the top right we have 90 degrees minus theta.
 The mystery angle x is 180 degrees. So theta + 90 degreestheta + x = 180 degrees. Adding theta and negative theta gives us zero on the left, and subtracting 90 degrees from both sides leaves x equal to 90 degrees. So we have established that the mystery angle x = 90 degrees.

5Look at the whole figure as a trapezoid in two ways. First, the formula for a trapezoid is A= the Height x (Base1 + Base 2)/2. The height is a+b and (Base1 + Base 2)/2 = 1/2(a + b). So that all equals 1/2 (a+b)^2.

6Look at the interior of the trapezoid and add up the areas, in order to set them equal to the formula just found. We have the two smaller triangles at bottom and left, and those together equal 2*1/2(a*b), which just equals (a*b). Then we also have 1/2 c*c, or 1/2 c^2. So together we have the other formula for the area of the trapezoid equaling (a*b)+ 1/2 c^2.

7Set the two Area formulas equal. 1/2(a+b)^2=(a*b)+1/2 c^2. Now multiply both sides by 2 to get rid of the 1/2's 2(1/2 (a+b)^2) = 2((a*b)+ 1/2 c^2.) which simplifies as (a+b)^2 = 2ab + c^2.Advertisement
Part 2
Part 2 of 3:Explanatory Charts, Diagrams, Photos
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Part 3
Part 3 of 3:Helpful Guidance
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Part 3

1Make use of helper articles when proceeding through this tutorial:
 See the article Create Higher Exponential Powers Geometrically for a list of articles related to Excel, Geometric and/or Trigonometric Art, Charting/Diagramming and Algebraic Formulation.
 For more art charts and graphs, you might also want to click on Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category:Graphics to view many Excel worksheets and charts where Trigonometry, Geometry and Calculus have been turned into Art, or simply click on the category as appears in the upper right white portion of this page, or at the bottom left of the page.
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QuestionHow is sum of three angles of triangle 180?Community AnswerA square's angles add up to 360 degrees. A triangle is half of a square. If you divide 360 by 2, you get 180.
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 There are over 100 proofs of the Pythagorean Theorem  maybe you can find a new one!Thanks!
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References
 http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D47  resource source, content is shared on a Creative Commons Sharealike 3.0 license, Please refer esp. to Book I, Proposition 47
 Video: "James Garfield's proof of the Pythagorean Theorem." khanacademy Published on Nov 27, 2012 LICENSE: Creative Commons (AttributionNoncommercialNo Derivative Works).
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