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In geometry, Fagnano's problem is an optimization problem that was first stated by Giovanni Fagnano in 1775: For a given acute triangle determine the inscribed triangle of minimal perimeter . The solution is the orthic triangle , with vertices at the base points of the altitudes of the given triangle.
The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears that the area should be S = 13×5 / 2 = 32.5 units. 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.
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.
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden.
In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is = /, where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a ...
In this example, the triangle's side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well when the side lengths are real numbers. As long as they obey the strict triangle inequality, they define a triangle in the Euclidean plane whose area is a positive real number.
On a sphere or a hyperboloid, the area of a geodesic triangle, i.e. a triangle all the sides of which are geodesics, is proportional to the difference of the sum of the interior angles and π. The constant of proportionality is just the Gaussian curvature, a constant for these surfaces.
A magic triangle or perimeter magic triangle [1] is an arrangement of the integers from 1 to n on the sides of a triangle with the same number of integers on each side, called the order of the triangle, so that the sum of integers on each side is a constant, the magic sum of the triangle.