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In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem [1] or the upside down Pythagorean theorem [2]) is as follows: [3] Let A, B be the endpoints of the hypotenuse of a right triangle ABC. Let D be the foot of a perpendicular dropped from C, the vertex of the right angle, to the hypotenuse. Then
In a right triangle, the sum of the reciprocals of the squares of the altitudes from the legs (equivalently, of the squares of the legs themselves) equals the reciprocal of the square of the altitude from the hypotenuse (the inverse Pythagorean theorem).
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.
By the Pythagorean theorem we have b 2 = h 2 + d 2 and a 2 = h 2 + (c − d) 2 according to the figure at the right. Subtracting these yields a 2 − b 2 = c 2 − 2cd. This equation allows us to express d in terms of the sides of the triangle: = + +.
Illustration of the inverse Pythagorean and regular Pythagorean theorems. The inverse Pythagorean theorem is obtained from the above equation by substituting x with AC, y with BC, and each a and b with CD, where A, B are the endpoints of the hypotenuse of a right triangle ABC, and D is the foot of a perpendicular dropped from C, the vertex of ...
The optic equation with squares appears in the inverse Pythagorean theorem (red) The optic equation, permitting but not requiring integer solutions, appears in several contexts in geometry . In a bicentric quadrilateral , the inradius r , the circumradius R , and the distance x between the incenter and the circumcenter are related by Fuss ...
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length , then applying the Pythagorean theorem and definitions of the
= (Geometric mean theorem; see Special Cases, inverse Pythagorean theorem) In a right triangle, the altitude from each acute angle coincides with a leg and intersects the opposite side at (has its foot at) the right-angled vertex, which is the orthocenter.