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For two of these, the external tangent lines, the circles fall on the same side of the line; for the two others, the internal tangent lines, the circles fall on opposite sides of the line. The external tangent lines intersect in the external homothetic center, whereas the internal tangent lines intersect at the internal homothetic center. Both ...
Figure 9: The two tangent lines of the two tangent points of a given circle intersect on the radical axis R (red line) of the two solution circles (pink). The three points of intersection on R are the poles of the lines connecting the blue tangent points in each given circle (black). Gergonne's approach is to consider the solution circles in ...
Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans (Latin for "kissing circle") by Leibniz.
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.
[4]: p.42 The line containing them is the Newton line of the quadrilateral. If the extensions of opposite sides in a tangential quadrilateral intersect at J and K, and the extensions of opposite sides in its contact quadrilateral intersect at L and M, then the four points J, L, K and M are collinear. [20]: Cor.3
If a point P moves along a line l, its polar p rotates about the pole L of the line l. If two tangent lines can be drawn from a pole to the circle, then its polar passes through both tangent points. If a point lies on the circle, its polar is the tangent through this point. If a point P lies on its own polar line, then P is on the circle.
Harcourt's theorem is a formula in geometry for the area of a triangle, as a function of its side lengths and the perpendicular distances of its vertices from an arbitrary line tangent to its incircle. [1] The theorem is named after J. Harcourt, an Irish professor. [2]
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