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2. Monge's theorem - Wikipedia

   en.wikipedia.org/wiki/Monge's_theorem

   The three spheres can be sandwiched uniquely between two planes. Each pair of spheres defines a cone that is externally tangent to both spheres, and the apex of this cone corresponds to the intersection point of the two external tangents, i.e., the external homothetic center. Since one line of the cone lies in each plane, the apex of each cone ...

3. 3-sphere - Wikipedia

   en.wikipedia.org/wiki/3-sphere

   Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres) In mathematics, a hypersphere, 3-sphere, or glome is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central ...

4. Overlapping circles grid - Wikipedia

   en.wikipedia.org/wiki/Overlapping_circles_grid

   The number of circles is n 3-(n-1) 3 = 3n 2-3n+1 = 3n(n-1)+1. These overlapping circles can also be seen as a projection of an n-unit cube of spheres in 3-dimensional space, viewed on the diagonal axis. There are more spheres than circles because some are overlapping in 2 dimensions.

5. Intersection (geometry) - Wikipedia

   en.wikipedia.org/wiki/Intersection_(geometry)

   There are two possibilities: if =, the spheres coincide, and the intersection is the entire sphere; if , the spheres are disjoint and the intersection is empty. When a is nonzero, the intersection lies in a vertical plane with this x-coordinate, which may intersect both of the spheres, be tangent to both spheres, or external to both spheres.

6. Trilateration - Wikipedia

   en.wikipedia.org/wiki/Trilateration

   Trilateration in three-dimensional geometry Intersection point of three pseudo-ranges. Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth (geopositioning). [1] When more than three distances are involved, it may be called multilateration, for emphasis.

7. Cher Says She Once Considered Jumping Off a Balcony ... - AOL

   www.aol.com/cher-says-she-once-considered...

   During an appearance on The Tonight Show Starring Jimmy Fallon in November 2023, Cher said that her life was "too immense for only one book." On Nov. 20, Cher will embark on a book tour that's ...

8. Miquel's theorem - Wikipedia

   en.wikipedia.org/wiki/Miquel's_theorem

   Draw three circumcircles (Miquel's circles) to triangles AB´C´, A´BC´, and A´B´C. Miquel's theorem states that these circles intersect in a single point M, called the Miquel point. In addition, the three angles MA´B, MB´C and MC´A (green in the diagram) are all equal, as are the three supplementary angles MA´C, MB´A and MC´B. [2] [3]

9. Judge overseeing 'fake elector' case recuses himself after ...

   www.aol.com/judge-overseeing-fake-elector-case...

   The judge overseeing the Arizona "fake elector" case against several of Donald Trump's allies has recused himself from the case after accusations of personal bias. Last week, defense attorneys for ...