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In an equilateral triangle the area of the Malfatti circles (left) is approximately 1% smaller than the three area-maximizing circles (right).. Gian Francesco Malfatti () posed the problem of cutting three cylindrical columns out of a triangular prism of marble, maximizing the total volume of the columns.
The equilateral triangle can be constructed in different ways by using circles. The first proposition in the Elements first book by Euclid . Start by drawing a circle with a certain radius, placing the point of the compass on the circle, and drawing another circle with the same radius; the two circles will intersect in two points.
The nine-point circle is tangent to the incircle and excircles. In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: [28] [29] The midpoint of each side of the triangle; The foot ...
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. ... In the case of the equilateral triangle, ...
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
Circle packing in an equilateral triangle is a packing problem in discrete mathematics where the objective is to pack n unit circles into the smallest possible equilateral triangle. Optimal solutions are known for n < 13 and for any triangular number of circles, and conjectures are available for n < 28. [1] [2] [3]
Euler's inequality, in the form stating that, for all triangles inscribed in a given circle, the maximum of the radius of the inscribed circle is reached for the equilateral triangle and only for it, is valid in absolute geometry. [7]
For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other.