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If a Jordan curve is inscribed in an annulus whose outer radius is at most + times its inner radius, and it is drawn in such a way that it separates the inner circle of the annulus from the outer circle, then it contains an inscribed square. In this case, if the given curve is approximated by some well-behaved curve, then any large squares that ...
Dante's image also calls to mind a passage from Vitruvius, famously illustrated later in Leonardo da Vinci's Vitruvian Man, of a man simultaneously inscribed in a circle and a square. [48] Dante uses the circle as a symbol for God, and may have mentioned this combination of shapes in reference to the simultaneous divine and human nature of Jesus.
A polygon inscribed in a circle, ellipse, or polygon (or a polyhedron inscribed in a sphere, ellipsoid, or polyhedron) has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure.
For instance, a square can be inscribed on any circle, which becomes its circumscribed circle. As another special case of the inscribed square problem, a square can be inscribed on the boundary of any convex set. The only other regular polygon with this property is the equilateral triangle. More strongly, there exists a convex set on which no ...
[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.
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 ...
Inscribe a square in the circle, so that its four corners lie on the circle. Between the square and the circle are four segments. If the total area of those gaps, G 4, is greater than E, split each arc in half. This makes the inscribed square into an inscribed octagon, and produces eight segments with a smaller total gap, G 8.
Any square can be inscribed in a circle whose center is the center of the square. If the common length of its four sides is equal to a {\displaystyle a} then the length of the diagonal is equal to a 2 {\displaystyle a{\sqrt {2}}} according to the Pythagorean theorem , and Ptolemy's relation obviously holds.