Lesson HOW TO construct a common interior tangent line to two circles

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HOW TO construct a common interior tangent line to two circles

In this lesson you will learn how to construct a common interior tangent line to two circles in a plane located each outside the other using a ruler and a compass.

Problem 1

For two given circles in a plane located each outside the other, to construct the common interior tangent line using a ruler and a compass.

Solution

We are given two circles in a plane located each outside the other (Figure 1a).
We need to construct the common interior tangent line to the circles using a ruler and a compass.

First, let us analyze the problem and make a sketch (Figures 1a and 1b). Let AB be the common interior tangent line to the circles we are searching for.
Let us connect the tangent point A of the first circle with its center P and the tangent point B of the second circle with its center Q (Figure 1a).

Then the radii PA and QB are both perpendicular to the tangent line AB (lesson A tangent line to a circle is perpendicular to the radius drawn to the tangent point under the topic Circles and their properties of the section Geometry in this site). Hence, the radii PA and QB are parallel.

Figure 1a. To the Problem 1

Figure 1b. To the solution of the Problem 1

Next, let us draw the straight line PQ connecting the centers P and Q, and let M be the intersection point of the straight line PQ with the common interior
tangent line AB (Figure 1b). The triangles DELTA

PAM and DELTA

QBM are similar, because they are right-angled triangles and have congruent vertical angles LPMA
and LQMB. It implies proportionality of the side lengths of the triangles:
abs%28PM%29%2Fabs%28QM%29

.

This means that the procedure of constructing the common exterior tangent line to two circles should be as follows:
1) subdivide the segment PQ in proportion r%5B1%5D%2Fr%5B2%5D

counting from the point P. It will give you the point M on the segment PQ (Figure 1b).
         It is described in the lesson HOW TO construct the segment whose length is an unknown term of a proportion how to divide a given segment in a given proportion.

    2) construct the tangent line to the first circle from the point M (the segment MA in Figure 1b). The method of constructing such a tangent line is described
         in the lesson HOW TO construct a tangent line to a circle through a given point outside the circle under the current topic.
         In this way you will get the tangent point A on the first circle of the radius r%5B1%5D

;
3) construct the tangent line to the second circle from the point M (the segment MB in Figure 1b). Use the same method of constructing such a tangent line
as in the previous step. In this way you will get the tangent point B on the second circle of the radius r%5B2%5D

;
4) The three points A, M and B lie in one straight line which is the required common interior tangent line to the two given circles.

Note that all these operations 1) - 3) can be done using a ruler and a compass. The problem is solved.

Problem 2

Find the length of the common exterior tangent segment to two given circles in a plane, if they have the radii r%5B1%5D

and

and the distance between their centers is d.
The circles are located each outside the other.

Solution

Let us use the Figure 1b from the solution to the previous Problem 1. This Figure is relevant to the Problem 2. It is copied and reproduced in the Figure 2 for your convenience.

It is clear from the Figure 2 and from the solution of the Problem 1
above that the length of the common interior tangent segment |AB| is
equal to the sum of the lengths of the segments |AM| and |MM|:

|AB| = |AM| + |BM|. (1)

From the other side, the triangles DELTA

PAM and DELTA

QBM are similar,
and it implies proportionality of the side lengths:
abs%28PM%29%2Fabs%28QM%29

. (2)

Figure 2. To the solution of the Problem 2

In addition,

|PM| + |QM| = d. (3)

The relations (2) and (3) imply

|PM| = %28d%2Ar%5B1%5D%29%2F%28r%5B1%5D+%2B+r%5B2%5D%29

, |QM| = %28d%2Ar%5B2%5D%29%2F%28r%5B1%5D+%2B+r%5B2%5D%29

.
Hence,
|AB| = |AM| + |BM| = sqrt%28abs%28PM%29%5E2-abs%28PA%29%5E2%29

sqrt%28abs%28QM%29%5E2-abs%28QB%29%5E2%29

sqrt%28%28%28d%2Ar%5B1%5D%29%2F%28r%5B1%5D+%2B+r%5B2%5D%29%29%5E2+-+r%5B1%5D%5E2%29

sqrt%28%28%28d%2Ar%5B2%5D%29%2F%28r%5B1%5D+%2B+r%5B2%5D%29%29%5E2+-+r%5B2%5D%5E2%29

.
The problem is solved.

Example 1

Find the length of the common interior tangent segment to two given circles in a plane, if their radii are 3 cm and 1.8 cm and the distance between their centers
is 8 cm.

Solution

Use the Figure 2 and the formulas derived in the solution of the Problem 2 above.
According to this formula, the length of the segment AM is equal to

sqrt%28%28%28d%2Ar%5B1%5D%29%2F%28r%5B1%5D+%2B+r%5B2%5D%29%29%5E2+-+r%5B1%5D%5E2%29

sqrt%28%28%288%2A3%29%2F%283+%2B+1.8%29%29%5E2+-+3%5E2%29

= 4 cm;

the length of the segment BM is equal to

sqrt%28%28%28d%2Ar%5B2%5D%29%2F%28r%5B1%5D+%2B+r%5B2%5D%29%29%5E2+-+r%5B1%5D%5E2%29

sqrt%28%28%288%2A1.8%29%2F%283+%2B+1.8%29%29%5E2+-+1.8%5E2%29

sqrt%28%2814.4%2F4.8%29%5E2+-+1.8%5E2%29

= 2.4 cm.

Hence, the length of the common interior tangent segment is 4 cm + 2.4 cm = 6.4 cm.

Answer. The length of the common interior tangent segment to the two given circles is 6.4 cm.

My other lessons on circles in this site, in the logical order, are
    - A circle, its chords, tangent and secant lines - the major definitions,
    - 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 topic Circles and their properties of the section Geometry, 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,
    - 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 current topic.

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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