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In general, the same inversion transforms the given line L and given circle C into two new circles, c 1 and c 2. Thus, the problem becomes that of finding a solution line tangent to the two inverted circles, which was solved above. There are four such lines, and re-inversion transforms them into the four solution circles of the Apollonius problem.
The line segments OT 1 and OT 2 are radii of the circle C; since both are inscribed in a semicircle, they are perpendicular to the line segments PT 1 and PT 2, respectively. But only a tangent line is perpendicular to the radial line. Hence, the two lines from P and passing through T 1 and T 2 are tangent to the circle C.
To show this, Gergonne considered lines through corresponding points of tangency on two of the given circles, e.g., the line defined by A 1 /A 2 and the line defined by B 1 /B 2. Let X 3 be a center of similitude for the two circles C 1 and C 2; then, A 1 /A 2 and B 1 /B 2 are pairs of antihomologous points, and their lines intersect at X 3. It ...
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 ...
The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th
For an internally tangent circle that circumscribes the other circles, the sign is negative. If a straight line is considered a degenerate circle with zero curvature (and thus infinite radius), Descartes' theorem also applies to a line and three circles that are all three mutually tangent (see Generalized circle). [1]
a) different tangent lines (transversal intersection, after transversality), or b) the tangent line in common and they are crossing each other (touching intersection, after tangency). If both the curves have a point S and the tangent line there in common but do not cross each other, they are just touching at point S.
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