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Configurations of Lines and Circles#

Theorem: Line and Circle Intersections

There are three possible cases for a [[Straight Line]] and a [[Circle]]:

  • They have no points of intersections.
  • They intersect at one point.
  • They intersect at two points.

![[res/Circle and Line Intersections.svg]]

Proof

Suppose, towards contradiction, that the [[Straight Line]] \(p\) and the [[Circle]] \(k(O; r)\) intersect at three points.

TODO

External Lines#

Definition: External Line

Let \(p\) be a [[Straight Line]] and let \(k(O; r)\) be a [[Circle#Circle|circle]] such that all points of [[Circle#Circle|circle]]

If all points of \(p\) lie outside of \(k\), then \(p\) is known as an external line for \(k\).

Theorem: External Lines in a Plane

If a [[Straight Line]] \(p\) and a [[Circle#Circle|circle]] \(k(O; r)\) lie in the same [[Planes|plane]] and do not intersect, then \(p\) is an [[Configurations of Lines and Circles|external line]] for \(k\).

\[ d(O; P) \gt r \qquad \forall P \in p \]

![[res/External Line of Circle.svg]]

Proof

TODO

Tangent Lines#

Definition: Tangent Line

A [[Straight Line]] is a tangent line to a [[Circle]] iff they have only one point of intersection.

![[res/Tangent Line to Circle.svg]]

Characterizations#

Theorem: Perpendicularity of Tangent Lines

Let \(k(O;r)\) be a [[Circle#Circle|circle]] and let \(p\) be a [[Straight Line]] which intersects \(k\) at a point \(P\).

The line \(p\) is a [[Configurations of Lines and Circles#Tangent Lines|tangent line]] if and only if the [[Circle#Circle|radius]] from \(O\) to \(P\) is [[TODO|perpendicular]] to \(p\).

![[res/Perpendicularity of Tangent Lines and Circles.svg]]

Proof

TODO

Properties#

Theorem

If \(k(O; r)\) is a [[Circle#Circle|circle]] and \(p\) is a [[Configurations of Lines and Circles#Tangent Lines|tangent line]], then all points of \(p\), apart from the point of intersection with \(k\), lie outside of \(k\).

Proof

TODO

Theorem: Tangent Lines through External Points

Let \(k(O; r)\) be a [[Circle#Circle|circle]].

If \(P\) is a point outside of \(k\) but which lies in the [[Planes|plane]] of \(k\), then there exist precisely two [[Configurations of Lines and Circles#Tangent Lines|tangent lines]] to \(k\) which pass through \(P\). Moreover, the [[Triangles]] formed by \(P\), \(O\) and the points of intersection are [[Triangles|congruent]].

\[ \triangle PTO \cong \triangle PT'O \]

![[res/Tangents to Circle Through External Point.svg]]

Proof

TODO

Secant Lines#

Definition: Secant Line

A [[Straight Line]] is a secant line to a [[Circle#Circle|circle]] iff they intersect at two points.

![[res/Secant Line through Circle.svg]]

Properties#

Intersecting Secants Theorem

Let \(k(O; r)\) be a [[Circle#Circle|circle]].

If \(s_1\) and \(s_2\) are two [[Configurations of Lines and Circles#Secant Lines|secant lines]] such that \(s_1\) intersects \(k\) at the points \(A\) and \(A'\), \(s_2\) intersects \(k\) at the points \(B\) and \(B'\), and the two secant lines intersect each other at \(C\), then the [[Triangles]] \(\triangle CAB'\) and \(\triangle CBA'\) are [[TODO|similar]].

\[ \triangle CAB' \sim \triangle CBA' \]

![[res/Intersecting Secants Theorem.svg]]

Proof

The [[Angles#Inscribed Angles|inscribed angles]] \(\angle AB'B\) and \(\angle BA'A\) are equal, since they have the same [[Angles#Inscribed Angles|corresponding arc]], namely \(\overset{\frown}{AB}\). The angle \(\angle A'CB'\) is shared by both triangles. The third angles of the triangles are then equal because the sum of the angles in a triangle is always \(180\degree\). Since the triangles have equal

Bibliography#