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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.
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 ...
k = 1 is the tangent line to the right of the circles looking from c 1 to c 2. k = −1 is the tangent line to the right of the circles looking from c 2 to c 1. The above assumes each circle has positive radius. If r 1 is positive and r 2 negative then c 1 will lie to the left of each line and c 2 to the right, and the two tangent lines will ...
For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In the three-dimensional case a surface normal , or simply normal , to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P .
[1] An excircle or escribed circle [2] of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. [3]
It's a classic tale: You have last-minute guests coming over for dinner or a bake sale fundraiser you didn't find out about until the night before—and now you need to concoct some tasty treats ...
[1] [2] More precisely, a straight line is tangent to the curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f ' (c), where f ' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
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