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In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs. Since the tangent line to a circle at a point P is ...
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.
Kissing circles. Given three mutually tangent circles (black), there are, in general, two possible answers (red) as to what radius a fourth tangent circle can have. In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this ...
If a tangent from an external point A meets the circle at F and a secant from the external point A meets the circle at C and D respectively, then AF 2 = AC × AD (tangent–secant theorem). The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side ...
Secant-, chord-theorem. For the intersecting secants theorem and chord theorem the power of a point plays the role of an invariant: . Intersecting secants theorem: For a point outside a circle and the intersection points , of a secant line with the following statement is true: | | | | = (), hence the product is independent of line .
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 ...
If , are tangent from different sides of (one in and one out), is the length of the interior common tangent. The converse of Casey's theorem is also true. [4] That is, if equality holds, the circles are tangent to a common circle.
Then, the image of the -excircle under is a circle internally tangent to sides , and the circumcircle of , that is, the -mixtilinear incircle. Therefore, the A {\displaystyle A} -mixtilinear incircle exists and is unique, and a similar argument can prove the same for the mixtilinear incircles corresponding to B {\displaystyle B} and C ...