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In elementary geometry the word congruent is often used as follows. [2] The word equal is often used in place of congruent for these objects. Two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure. Two circles are congruent if they have the same diameter.
Descartes' theorem generalizes to mutually tangent great or small circles in spherical geometry if the curvature of the th circle is defined as = , the geodesic curvature of the circle relative to the sphere, which equals the cotangent of the oriented intrinsic radius.
In addition, any two non-intersecting circles may be inverted into congruent circles, using circle of inversion centered at a point on the circle of antisimilitude. The Peaucellier–Lipkin linkage is a mechanical implementation of inversion in a circle. It provides an exact solution to the important problem of converting between linear and ...
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle. Table of solutions, 1 ≤ n ≤ 20 [ edit ]
In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems , and play an important role in many geometrical constructions and proofs .
The nine-point circles are all congruent with a radius of half that of the cyclic quadrilateral's circumcircle. The nine-point circles form a set of four Johnson circles. Consequently, the four nine-point centers are cyclic and lie on a circle congruent to the four nine-point circles that is centered at the anticenter of the cyclic quadrilateral.
The reference triangle is in fact congruent to the Johnson triangle, and is homothetic to it by a factor −1. The point H is the orthocenter of the reference triangle and the circumcenter of the Johnson triangle. The homothetic center of the Johnson triangle and the reference triangle is their common nine-point center.
There is an upper bound for radius of the inscribed circle. Two triangles are congruent if and only if they correspond under a finite product of line reflections. Two triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent).