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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 .
The Nagel triangle or extouch triangle of is denoted by the vertices , , and that are the three points where the excircles touch the reference and where is opposite of , etc. This T A T B T C {\displaystyle \triangle T_{A}T_{B}T_{C}} is also known as the extouch triangle of A B C {\displaystyle \triangle ABC} .
Each excircle touches one of these lines from the opposite side of the triangle, and is on the same side as the triangle for the other two lines. Like the incircle, the excircles are all tangent to the nine-point circle. Their points of tangency with the nine-point circle form a triangle, the Feuerbach triangle.
In plane geometry, the Conway circle theorem states that when the sides meeting at each vertex of a triangle are extended by the length of the opposite side, the six endpoints of the three resulting line segments lie on a circle whose centre is the incentre of the triangle. The circle on which these six points lie is called the Conway circle of ...
Determination of a circle, that intersects four circles by the same angle. [2] Solving the Problem of Apollonius; Construction of the Malfatti circles: [3] For a given triangle determine three circles, which touch each other and two sides of the triangle each. Spherical version of Malfatti's problem: [4] The triangle is a spherical one.
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]
Circle packing theorem, the result that every planar graph may be realized by a system of tangent circles; Hexafoil, the shape formed by a ring of six tangent circles; Feuerbach's theorem on the tangency of the nine-point circle of a triangle with its incircle and excircles; Descartes' theorem; Ford circle; Bankoff circle; Archimedes' twin circles
Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically .