Tangencies

The tangency , in planar geometry, is generally set between lines, circles, ellipses, parabolas or hyperbolas, with its different positions, sizes and combinations relative to each other.

Summary

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  • 1 Definition
  • 2 Basic properties of tangencies
  • 3 Nomenclature
  • 4 Simple tangencies
    • 1 Between a circumference and a line
    • 2 Between two circles
  • 5 Relative positions between two circles
  • 6 Some examples of multiple tangencies
    • 1 Tangent lines
    • 2 Tangent lines to two circles
    • 3 Tangent circumference
    • 4 Tangent circumference to another at a given point
    • 5 Tangent circumferences to other three given
  • 7 Sources

Definition

Two figures are said to be tangent when they have a single point in common, which is known as a point of tangency. The harmonic union between curves and straight lines or curves to each other is called bonding and this union must take place by tangency.

Tangencies can occur between circles, between circles and lines, between polygons and lines, between circles and polygons, etc … However, the most common tangencies in geometric drawings are those that are generated between lines and circles, and between circles each.

Basic properties of tangencies

To accurately solve tangent plots, the following theorems must be taken into account.

  • First theorem: a line is tangent to a circle when they have only one point (A) in common with each other, and the line is perpendicular to the radius of the circle at point (A).
  • Second theorem: a circle is tangent to two lines that intersect if its center is located on the bisector of the angle that the lines form.
  • Third theorem: two circles are tangent if they have a point in common aligned with the centers of the circle.

Nomenclature

  • Straight will refer to or indicate that straight lines are indefinite or infinitely long. You will also use ray (if you have one end) or segment (when you have two ends).
  • Circumference will refer to or indicate the fully closed. For cases where it is not closed, a circumference arc or simply an arc is used when clear.
  • Two circles, or a circle and a line, are tangent or tangent when they have a single point in common.
  • We will use color abbreviations, such as N = black (for the problem statement), A = blue (most representative steps), and R = red (for the solution).

Before drawing the tangent circumference to the given objects, mark the place of tangency lightly, either with a perpendicular to the tangent line passing through the center of the circumference, or otherwise joining with a line the centers of two circumferences extending it to the supposed place of tangency. This helps to draw arcs where needed.

Simple tangencies

They will be the most frequent tangencies and of easy resolution. They depend on the order in which the data is given for execution.

Between a circumference and a line

Objectively there are two main cases:

Given any circumference to which a tangent line is to be drawn:

  • In a first step it is necessary to identify and represent the diameter (A) of said circumference, which is perpendicular to the direction of the tangent line. Then it is a matter of drawing the line through the ends (R) of the found diameter or radius.

Given any line on which a tangent circle has to be constructed:

  • In a first step it is necessary to draw a perpendicular (A) on said line at the desired point of tangency. Then it is a matter of identifying the center of the circumference on said perpendicular (R) and drawing the circumference.

Between two circles

Given a circumference (N) to which we want to draw another tangent circumference:

  • In a first step it is necessary to draw a line that passes through the center of the given circumference and through the desired point of tangency (A). Then the center of the second circle (R) is identified to finish tracing it.

It is observed that the two centers of the circles and the place of tangency of these are always aligned.

Relative positions between two circles

The different adjectives are intuitive and will help to observe, identify and distinguish cases in which there are or are not tangencies and in which situation they occur. They are purely operational and informative, since in themselves they do not solve any problem, but they facilitate their explanation for complex cases.

Some examples of multiple tangencies

The most common cases of tangency are solved using methods such as bisectors , bisectors and longitudinal subtraction of radios, including methods homotecias or powers , to find circles tangent to two straight lines and a circle or also to find circles tangent to other three different sizes. Tangent Circumferences to Two Given Lines

  • The centers of the tangent circles are always points of the bisector of the angle formed by the two given lines.

There are two ways to find these centers:

  • On the one hand, drawing the bisector and then selecting the desired centers.
  • If you do not have access to the angle of the two given lines, the second way to find the bisector is to find two points of it, for example drawing equidistant parallels to both lines.

Tangent circumference to three given lines

  • The bisectors of the angles made up of two pairs of lines will be found.
  • The intersection of the two bisectors is the center of the circle we are looking for.

Tangent circumference to another at a given point and to a given line

  • A line tangent to the circumference is drawn at the given point.

From this line there are two possibilities:

  • First, the bisector of the two lines is drawn. The intersection with the bisector and the radial line is the center of this circumference.
  • Secondly, power can be applied, which in short, is equivalent to drawing a circle centered on the angle and radius to the point of tangency, said circle generates two points on the given line, which are the two possible tangencies of the circle sought. Finally with perpendiculars in said pair of points the centers of the sought circumferences are obtained.

Tangent lines

Tangent lines to a given circumference passing through a given outer point

  • We join by a segment the center of the circumference with the given outer point
  • We draw an auxiliary circumference whose diameter is the anterior segment.
  • The intersection of both circles are the points of tangency that when joined with the outer point give us the solution.

Tangency points are the vertices of the arc capable of 90º between the center of the circumference and the outer point.

Tangent lines to two circles

  • This problem can be converted into the previous one, by subtracting the length of the radius of the smaller circle from the two radii. The tangency points of the major circumference will be aligned with its center.
  • To find the interior tangents, add the length of the smallest radius to the largest and subtract the smallest. The tangency points of the major circumference will be aligned with its center.

Tangent circumference

Tangent circumference to another at a given point and passing through another given point

  • By means of a segment (A) the point of tangency of the circles is united with the given outer point.
  • The perpendicular bisector of this segment is made.
  • The center of the searched circle will be aligned with the center of the given circle and its tangent point.
  • Finally at the intersection is the center of the sought circumference.

Tangent circumference to another at a given point

Tangent circumference to another at a given point and to any other circumference

  • It is possible to deduce this method from the previous one.
  • By lengthening the radius of the circumference, with tangency, and on its point of tangency, we place two points that equidistant to that point the radius of the other circumference. Now identifying perpendicular bisectors between each of the points and the center of the circumference, which has no tangency, they generate two other points that are the centers of the two circles sought.

Tangent circumferences to three other given

In this case, it is only mentioned that several types of tangencies appear: an external tangent, three tangents each external to one and surrounding two others, three tangents each external to two and surrounding each other, and a tangent surrounding all three circumferences .

 

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