Lesson A circle, its chords, tangent and secant lines - the major definitions

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A circle, its chords, tangent and secant lines - the major definitions


Definitions


A circle is the set of all points in a plane that are located at the certain fixed distance from a given point in the plane called the  center  of the circle.

    In other words,  a circle is a locus of points in a plane that are  equidistant  from the center.


A  radius of the circle  is a straight line segment connecting the center of the circle with the point in the circle.

    One endpoint of a radius is the center of the circle. The other endpoint of the radius is a point in the circle.
    A circle has many radii that connect the center with different points in the circle.
    All these radii have the came length.  This common value of the radii length is also sometime called the  radius  of the circle.


A  chord  of a circle is a straight segment connecting two points in the circle.

    The endpoints of a chord are points in a circle.


A  diameter  of a circle is a chord that passes through its center.


A  secant line  to a circle is a straight line that intersects a circle in two points.


A  tangent line  to a circle is a straight line that intersects a circle exactly in one point.

Theorem 1

For any three given points in a plane that do not lie in one straight line there is a circle passing through these points,  and such a circle is unique.

Proof

Connect these three points by straight line segments.  You will get a triangle.
It was proved in the lesson  Perpendicular bisectors of a triangle sides are concurrent  in this site that for any triangle in a plane
there is a circle circumscribed around the triangle.
This circumscribed circle passes through its vertices that are the given points.
The center of this triangle is the intersection of the perpendicular bisectors of its sides.
Recall that the tree perpendicular bisectors of the tree sides of any triangle in a plane are concurrent.

Regarding the uniqueness, the center of the circle passing through three given points in the plane not lying in one straight line, is the intersection of the perpendicular
bisectors to the side of the triangles. This intersection (and thus the center of the circle) is uniquely defined.
And the radius of the circle is the distance from the intersection point of the tree perpendicular bisectors to any of the vertices of the triangle.
It is uniquely defined, too.

Corollary 1

The circle is uniquely defined by any three its distinct points.

Corollary 2

Two circles in a plane may have no more than two intersection points.


My other lessons on circles in this site are
    - The longer is the chord the larger its central angle is,
    - The chords of a circle and the radii perpendicular to the chords,
    - A tangent line to a circle is perpendicular to the radius drawn to the tangent point,
    - An inscribed angle in a circle,
    - Two parallel secants to a circle cut off congruent arcs,
    - The angle between two chords intersecting inside a circle,
    - The angle between two secants intersecting outside a circle,
    - The angle between a chord and a tangent line to a circle,
    - Tangent segments to a circle from a point outside the circle,
    - The converse theorem on inscribed angles,
    - The parts of chords that intersect inside a circle,
    - Metric relations for secants intersecting outside a circle  and
    - Metric relations for a tangent and a secant lines released from a point outside a circle
under the current topic,  and
    - HOW TO bisect an arc in a circle using a compass and a ruler,
    - HOW TO find the center of a circle given by two chords,
    - Solved problems on a radius and a tangent line to a circle,
    - Solved problems on inscribed angles,
    - A property of the angles of a quadrilateral inscribed in a circle,
    - An isosceles trapezoid can be inscribed in a circle,
    - HOW TO construct a tangent line to a circle at a given point on the circle,
    - HOW TO construct a tangent line to a circle through a given point outside the circle,
    - HOW TO construct a common exterior tangent line to two circles,
    - HOW TO construct a common interior tangent line to two circles,
    - Solved problems on chords that intersect within a circle,
    - Solved problems on secants that intersect outside a circle,
    - Solved problems on a tangent and a secant lines released from a point outside a circle
    - The radius of a circle inscribed into a right angled triangle
    - Solved problems on tangent lines released from a point outside a circle
under the topic  Geometry  of the section  Word problems.

The overview of lessons on Properties of Circles is in this file  PROPERTIES OF CIRCLES, THEIR CHORDS, SECANTS AND TANGENTS.
You can use the overview file or the list of links above to navigate over these lessons.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link  GEOMETRY - YOUR ONLINE TEXTBOOK.

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