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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.
The solution to the quadrilateral isoperimetric problem is the square, and the solution to the triangle problem is the equilateral triangle. In general, the polygon with n sides having the largest area and a given perimeter is the regular polygon, which is closer to being a circle than is any irregular polygon with the same number of sides.
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.
Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Applications requiring triangle solutions include geodesy, astronomy, construction, and navigation.
The three splitters concur at the Nagel point of the triangle. A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. So any cleaver, like any splitter, divides the triangle into two paths each of whose length equals the semiperimeter.
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined ...
Langley's Adventitious Angles Solution to Langley's 80-80-20 triangle problem. Langley's Adventitious Angles is a puzzle in which one must infer an angle in a geometric diagram from other given angles. It was posed by Edward Mann Langley in The Mathematical Gazette in 1922. [1] [2]
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