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Solutions to Apollonius's problem generally occur in pairs; for each solution circle, there is a conjugate solution circle (Figure 6). [1] One solution circle excludes the given circles that are enclosed by its conjugate solution, and vice versa.
CLP problems generally have 4 solutions. The solution of this special case is similar to that of the CPP Apollonius solution. Draw a circle centered on the given point P; since the solution circle must pass through P, inversion in this [clarification needed] circle transforms the solution circle
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible ... solutions for n = 2, 3, 4 ...
The solutions to this problem are sometimes called the circles of Apollonius. The Apollonian gasket —one of the first fractals ever described—is a set of mutually tangent circles, formed by solving Apollonius' problem iteratively.
To convert between these two formulations of the problem, the square side for unit circles will be = + /. The optimal packing of 15 circles in a square Optimal solutions have been proven for n ≤ 30. Packing circles in a rectangle; Packing circles in an isosceles right triangle - good estimates are known for n < 300.
The algebraic solutions do not distinguish between internal and external tangencies among the circles and the given triangle; if the problem is generalized to allow tangencies of either kind, then a given triangle will have 32 different solutions and conversely a triple of mutually tangent circles will be a solution for eight different ...
A circle containing one acre is cut by another whose center is on the circumference of the given circle, and the area common to both is one-half acre. Find the radius of the cutting circle. The solutions in both cases are non-trivial but yield to straightforward application of trigonometry, analytical geometry or integral calculus.
This problem is known as the primitive circle problem, as it involves searching for primitive solutions to the original circle problem. [9] It can be intuitively understood as the question of how many trees within a distance of r are visible in the Euclid's orchard , standing in the origin.