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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle.It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
For example, if one of the legs of a right angle has a length of 3 and the other has a length of 4, then their squares add up to 25 = 9 + 16 = 3 × 3 + 4 × 4. Since 25 is the square of the hypotenuse, the length of the hypotenuse is the square root of 25, that is, 5.
A congruent number is defined as the area of a right triangle with rational sides. Because every congruum can be obtained (using the parameterized solution) as the area of a Pythagorean triangle, it follows that every congruum is congruent. Conversely, every congruent number is a congruum multiplied by the square of a rational number. [7]
In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. The fraction of the triangle's area that is filled by the square is no ...
However, the blue triangle has a ratio of 5:2 (=2.5), while the red triangle has the ratio 8:3 (≈2.667), so the apparent combined hypotenuse in each figure is actually bent. With the bent hypotenuse, the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square.
Every triangle can be inscribed in an ellipse, called its Steiner circumellipse or simply its Steiner ellipse, whose center is the triangle's centroid. Every triangle has an infinitude of inscribed ellipses. One of them is a circle, and one of them is the Steiner inellipse which is tangent to the triangle at the midpoints of the sides.
The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above, the internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any spherical triangle, the sum is more than 180 ...
The spiral is started with an isosceles right triangle, with each leg having unit length.Another right triangle (which is the only automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3.