| If v 1, v 2, | In similar triangles, the ratios of the sides are the same regardless of the size of the triangles, and depend upon the angles |
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| The left green parallelogram has the same area as the left, blue portion of the bottom parallelogram because both have the same base b and height h | The upside-down Pythagorean theorem, Jennifer Richinick, The Mathematical Gazette, Vol |
Specifically, the square of the measure of an m-dimensional set of objects in one or more parallel m-dimensional in n-dimensional is equal to the sum of the squares of the measures of the projections of the object s onto all m-dimensional coordinate subspaces.
| Interactive Mathematics Miscellany and Puzzles | Some well-known examples are 3, 4, 5 and 5, 12, 13 |
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| " is an Old Babylonian clay tablet from concerning the computation of the sides of a rectangle given its area and diagonal | The dissection consists of dropping a perpendicular from the vertex of the right angle of the triangle to the hypotenuse, thus splitting the whole triangle into two parts |
it is not until Euclid that we find a logical sequence of general theorems with proper proofs.
1| This statement is illustrated in three dimensions by the tetrahedron in the figure | "Theorem 1 and Theorem 2" |
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| Instead of a square it uses a , which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in the diagram | Mathematics in Ancient Iraq: A Social History |
While Euclid's proof only applied to convex polygons, the theorem also applies to concave polygons and even to similar figures that have curved boundaries but still with part of a figure's boundary being the side of the original triangle.
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