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.

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 such that all points of 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 $k(O;r)$ lie in the same plane and do not intersect, then $p$ is an external line for $k$.

$$d(O;P)>r\forall P\in p$$

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.

Characterizations

Theorem: Perpendicularity of Tangent Lines

Let $k(O;r)$ be a circle and let $p$ be a straight line which intersects $k$ at a point $P$.

The line $p$ is a tangent line if and only if the radius from $O$ to $P$ is perpendicular to $p$.

PROOF

TODO

Properties

THEOREM

If $k(O;r)$ is a circle and $p$ is a 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.

If $P$ is a point outside of $k$ but which lies in the plane of $k$, then there exist precisely two tangent lines to $k$ which pass through $P$. Moreover, the triangles formed by $P$, $O$ and the points of intersection are congruent.

$$△PTO\cong △P{T}^{\prime }O$$

PROOF

TODO

Secant Lines

Definition: Secant Line

A straight line is a secant line to a circle iff they intersect at two points.

Properties

Intersecting Secants Theorem

Let $k(O;r)$ be a circle.

If $s_1$ and $s_2$ are two secant lines such that $s_1$ intersects $k$ at the points $A$ and $A^{\prime }$, $s_2$ intersects $k$ at the points $B$ and $B^{\prime }$, and the two secant lines intersect each other at $C$, then the triangles $△CA{B}^{\prime }$ and $△CB{A}^{\prime }$ are similar.

$$△CA{B}^{\prime }\sim △CB{A}^{\prime }$$

PROOF

The inscribed angles $\angle A{B}^{\prime }B$ and $\angle B{A}^{\prime }A$ are equal, since they have the same corresponding arc, namely $\stackrel{⌢}{\mathrm{AB}}$. The angle $\angle A^{\prime }C{B}^{\prime }$ 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 $180°$. Since the triangles have equal

Bibliography