Inscribed Square In A Triangle
In elementary geometry, an inscribed square in a triangle is a square whose four vertices all lie on a given triangle. By the pigeonhole principle, two of the square's vertices, and the edge between them, must lie on one of the sides of the triangle. For instance, for the Calabi triangle depicted, the square with horizontal and vertical sides is inscribed; the other two squares in the figure are not inscribed. This is a special case of the inscribed square problem asking for a square whose vertices lie on a simple closed curve. However, although the inscribed square problem remains unsolved in general, it is known to have a solution for every polygon and for every convex set, two special cases that both apply to triangles. Every acute triangle has three inscribed squares, one lying on each of its three sides. In a right triangle there are two inscribed squares, one touching the right angle of the triangle and the other lying on the opposite side. An obtuse triangle has only o ...
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Calabi Triangle
The Calabi triangle is a special triangle found by Eugenio Calabi and defined by its property of having three different placements for the largest square that it contains. It is an isosceles triangle which is obtuse triangle, obtuse with an irrational number, irrational but algebraic number, algebraic ratio between the lengths of its sides and its base. Definition Consider the largest square that can be placed in an arbitrary triangle. It may be that such a square could be positioned in the triangle in more than one way. If the largest such square can be positioned in three different ways, then the triangle is either an equilateral triangle or the Calabi triangle. Thus, the Calabi triangle may be defined as a triangle that is not equilateral and has three placements for its largest square. Shape The triangle is isosceles which has the same length of sides as . If the ratio of the base to either leg is , we can set that . Then we can consider the following three cases: ;case 1) i ...
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