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  2. Tangent lines to circles - Wikipedia

    en.wikipedia.org/wiki/Tangent_lines_to_circles

    An inversion in their tangent point with respect to a circle of appropriate radius transforms the two touching given circles into two parallel lines, and the third given circle into another circle. Thus, the solutions may be found by sliding a circle of constant radius between two parallel lines until it contacts the transformed third circle.

  3. Malfatti circles - Wikipedia

    en.wikipedia.org/wiki/Malfatti_circles

    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 ...

  4. Tangent circles - Wikipedia

    en.wikipedia.org/wiki/Tangent_circles

    In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency : internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as trilateration and maximizing the ...

  5. Special cases of Apollonius' problem - Wikipedia

    en.wikipedia.org/wiki/Special_cases_of_Apollonius...

    A circle is tangent to a point if it passes through the point, and tangent to a line if they intersect at a single point P or if the line is perpendicular to a radius drawn from the circle's center to P. Circles tangent to two given points must lie on the perpendicular bisector. Circles tangent to two given lines must lie on the angle bisector.

  6. Power of a point - Wikipedia

    en.wikipedia.org/wiki/Power_of_a_point

    Similar considerations generate the second tangent circle, that meets the given circles at the points , (see diagram). All tangent circles to the given circles can be found by varying line . Positions of the centers Circles tangent to two circles. If is the center and the radius of the circle, that is tangent to the given circles at the points ...

  7. Problem of Apollonius - Wikipedia

    en.wikipedia.org/wiki/Problem_of_Apollonius

    Consider a solution circle of radius r s and three given circles of radii r 1, r 2 and r 3. If the solution circle is externally tangent to all three given circles, the distances between the center of the solution circle and the centers of the given circles equal d 1 = r 1 + r s, d 2 = r 2 + r s and d 3 = r 3 + r s, respectively.

  8. Descartes' theorem - Wikipedia

    en.wikipedia.org/wiki/Descartes'_theorem

    An Apollonian gasket with integer curvatures, generated by four mutually tangent circles with curvatures −10 (the outer circle), 18, 23, and 27. When four tangent circles described by equation (2) all have integer curvatures, the alternative fourth circle described by the second solution to the equation must also have an integer curvature ...

  9. Contact (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Contact_(mathematics)

    A circle with 1st-order contact (tangent) A circle with 2nd-order contact (osculating) A circle with 3rd-order contact at a vertex of a curve. For each point S(t) on a smooth plane curve S, there is exactly one osculating circle, whose radius is the reciprocal of κ(t), the curvature of S at t.

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