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"The art of numbering proceeds from the circle and the square. The circle is derived from the square and the square from the rectangle (literally, the T-square or the carpenter's square). The rectangle originate from the fact that 9x9 = 81 (that is, the multiplication table or properties of numbers as such).
30° Narrow rhombus (Beige) with the same side-length as the green triangle; Trapezoid (half hexagon or 3 triangles) (Red) that can be matched with three of the green triangles; Regular Hexagon (6 triangles) (Yellow) that can be matched with six of the green triangles; Square (Orange) with the same side-length as the green triangle
An excircle or escribed circle [2] of the triangle is a circle lying outside the triangle, tangent to one of its sides, and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.
A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. This shows that the square of the n th triangular number is equal to the sum of the first n cube numbers. Also, the square of the n th triangular number is the same as the sum of the cubes of the integers 1 to n.
[25] [26] (Thus, for example, if a square is deformed into a rhombus it remains tangential, though to a smaller incircle). If one side is held in a fixed position, then as the quadrilateral is flexed, the incenter traces out a circle of radius / where a,b,c,d are the sides in sequence and s is the semiperimeter.
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid , circumcenter , incenter and orthocenter were familiar to the ancient Greeks , and can be obtained by simple constructions .
Each square covers an entire hemisphere and their vertices lie along a great circle. This is called a spherical square dihedron. The Schläfli symbol is {4,2}. Six squares can tile the sphere with 3 squares around each vertex and 120-degree internal angles. This is called a spherical cube. The Schläfli symbol is {4,3}.
Circle with square and octagon inscribed, showing area gap. Suppose that the area C enclosed by the circle is greater than the area T = cr/2 of the triangle. Let E denote the excess amount. Inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments.
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