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

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<H2>HOW TO construct a common interior tangent line to two circles</H2>

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.

<H3>Problem 1</H3>For two given circles in a plane located each outside the other, &nbsp;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 &nbsp;(Figure 1a).

We need to construct the common interior tangent line to the circles using a ruler and a compass.

First, &nbsp;let us &nbsp;analyze&nbsp; the problem and make a sketch &nbsp;(Figures 1a&nbsp; and &nbsp;1b). &nbsp;Let &nbsp;AB&nbsp; be the common interior tangent line to the circles we are searching for.

Let us connect the tangent point &nbsp;A&nbsp; of the first circle with its center &nbsp;P&nbsp; and the tangent point &nbsp;B&nbsp; of the second circle with its center &nbsp;Q&nbsp; (Figure 1a).

Then the radii &nbsp;PA&nbsp; and &nbsp;QB&nbsp; are both perpendicular to the tangent line &nbsp;AB &nbsp;(lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Circles/A-tangent-line-to-a-circle-is-perpendicular-to-the-radius-drawn-to-the-tangent-point.lesson>A tangent line to a circle is perpendicular to the radius drawn to the tangent point</A>&nbsp; under the topic &nbsp;Circles and their properties&nbsp; of the section &nbsp;Geometry&nbsp; in this site). &nbsp;Hence, &nbsp;the radii &nbsp;PA&nbsp; and &nbsp;QB&nbsp; are parallel.

&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 420, 240,  -3.5, 10.5, -4.0, 4.0, 
            circle( 0.0, 0.0, 3.0), 
       blue(line( 0.0, 0.0, 1.8, 2.4)),
      green(line( 1.0, 3.0, 5.0, 0.0)),

locate(-0.2,  0.0, P),
            locate( 1.7,  2.85, A),

arc( 1.8, 2.4, 0.9, 0.9, 34, 123),
             arc( 1.8, 2.4, 1.2, 1.2, 34, 123),

circle( 8.0, 0.0, 1.8),
            locate( 8.0, 0.0, Q),

green(line( 5.0, 0.0, 7.72, -1.94)),

blue(line( 8.0, 0.0, 6.92, -1.44)),
            locate( 6.7, -1.5, B),

arc( 6.92, -1.44, 0.9, 0.9, 218, 304),
             arc( 6.92, -1.44, 1.2, 1.2, 218, 304)
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 1a. &nbsp;To the Problem 1

</TD>
 <TD>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 420, 240,  -3.5, 10.5, -4.0, 4.0, 
            circle( 0.0, 0.0, 3.0), 
            line(-3.5, 0.0, 10.3, 0.0),
       blue(line( 0.0, 0.0, 1.8, 2.4)),
      green(line( 1.0, 3.0, 5.0, 0.0)),

locate(-0.2,  0.0, P),
            locate( 1.7,  2.85, A),
            locate( 4.9, -0.1, M),

arc( 1.8, 2.4, 0.9, 0.9, 34, 123),
             arc( 1.8, 2.4, 1.2, 1.2, 34, 123),

circle( 8.0, 0.0, 1.8),
            locate( 8.0, 0.0, Q),

green(line( 5.0, 0.0, 7.72, -1.94)),

blue(line( 8.0, 0.0, 6.92, -1.44)),
            locate( 6.7, -1.5, B),

arc( 6.92, -1.44, 0.9, 0.9, 218, 304),
             arc( 6.92, -1.44, 1.2, 1.2, 218, 304)
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 1b. &nbsp;To the solution of the Problem 1

</TD>
 </TR>
</TABLE>

Next, &nbsp;let us draw the straight line &nbsp;PQ&nbsp; connecting the centers &nbsp;P&nbsp; and &nbsp;Q, &nbsp;and let &nbsp;M&nbsp; be the intersection point of the straight line &nbsp;PQ&nbsp; with the common interior

tangent line &nbsp;AB&nbsp; (Figure 1b). &nbsp;The triangles &nbsp;{{{DELTA}}}PAM&nbsp; and &nbsp;{{{DELTA}}}QBM&nbsp; are similar, &nbsp;because they are right-angled triangles and have congruent vertical angles &nbsp;LPMA

and &nbsp;LQMB. &nbsp;It implies proportionality of the side lengths of the triangles:

{{{abs(PM)/abs(QM)}}} = {{{abs(PA)/abs(QB)}}} = {{{r[1]/r[2]}}}.

This means that the procedure of constructing the common exterior tangent line to two circles should be as follows:

&nbsp;&nbsp;&nbsp;&nbsp;1) &nbsp;subdivide the segment &nbsp;PQ&nbsp; in proportion {{{r[1]/r[2]}}} counting from the point &nbsp;P. &nbsp;It will give you the point &nbsp;M&nbsp; on the segment &nbsp;PQ&nbsp; (Figure 1b).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is described in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Parallelograms/HOW-TO-construct-the-segment-whose-length-is-an-unknown-term-of-a-proportion.lesson>HOW TO construct the segment whose length is an unknown term of a proportion</A>&nbsp; how to divide a given segment in a given proportion.

&nbsp;&nbsp;&nbsp;&nbsp;2) &nbsp;construct the tangent line to the first circle from the point &nbsp;M&nbsp; &nbsp;(the segment &nbsp;MA&nbsp; in &nbsp;Figure 1b). &nbsp;The method of constructing such a tangent line is described

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-tangent-line-to-a-circle-through-a-given-point-outside-the-circle.lesson>HOW TO construct a tangent line to a circle through a given point outside the circle</A>&nbsp; under the current topic.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In this way you will get the tangent point &nbsp;A&nbsp; on the first circle of the radius &nbsp;{{{r[1]}}};

&nbsp;&nbsp;&nbsp;&nbsp;3) &nbsp;construct the tangent line to the second circle from the point &nbsp;M&nbsp; &nbsp;(the segment &nbsp;MB&nbsp; in &nbsp;Figure 1b). &nbsp;Use the same method of constructing such a tangent line

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;as in the previous step. &nbsp;In this way you will get the tangent point &nbsp;B&nbsp; on the second circle of the radius &nbsp;{{{r[2]}}};

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

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

<H3>Problem 2</H3>Find the length of the common exterior tangent segment to two given circles in a plane, &nbsp;if they have the radii {{{r[1]}}}&nbsp; and {{{r[2]}}}&nbsp; and the distance between their centers is &nbsp;d.

The circles are located each outside the other.

Solution

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

<TABLE> 
 <TR>
 <TD>
It is clear from the &nbsp;Figure 2&nbsp; and from the solution of the &nbsp;Problem 1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

above that the length of the common interior tangent segment &nbsp;|AB|&nbsp; is

equal to the sum of the lengths of the segments &nbsp;|AM|&nbsp; and &nbsp;|MM|:

&nbsp;|AB| = |AM| + |BM|.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

From the other side, &nbsp;the triangles &nbsp;{{{DELTA}}}PAM&nbsp; and &nbsp;{{{DELTA}}}QBM&nbsp; are similar,

and it implies proportionality of the side lengths:

{{{abs(PM)/abs(QM)}}} = {{{abs(PA)/abs(QB)}}} = {{{r[1]/r[2]}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)

</TD> 
 <TD>

{{{drawing( 420, 240,  -3.5, 10.5, -4.0, 4.0, 
            circle( 0.0, 0.0, 3.0), 
            line(-3.5, 0.0, 10.3, 0.0),
       blue(line( 0.0, 0.0, 1.8, 2.4)),
      green(line( 1.0, 3.0, 5.0, 0.0)),

locate(-0.2,  0.0, P),
            locate( 1.7,  2.85, A),
            locate( 4.9, -0.1, M),

arc( 1.8, 2.4, 0.9, 0.9, 34, 123),
             arc( 1.8, 2.4, 1.2, 1.2, 34, 123),

circle( 8.0, 0.0, 1.8),
            locate( 8.0, 0.0, Q),

green(line( 5.0, 0.0, 7.72, -1.94)),

blue(line( 8.0, 0.0, 6.92, -1.44)),
            locate( 6.7, -1.5, B),

arc( 6.92, -1.44, 0.9, 0.9, 218, 304),
             arc( 6.92, -1.44, 1.2, 1.2, 218, 304)
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Figure 2. &nbsp;To the solution of the Problem 2

</TD>
 </TR>
</TABLE>In addition,

|PM| + |QM| = d.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3)

The relations &nbsp;(2)&nbsp; and &nbsp;(3)&nbsp; imply

|PM| = {{{(d*r[1])/(r[1] + r[2])}}}, &nbsp;&nbsp;|QM| = {{{(d*r[2])/(r[1] + r[2])}}}.

Hence,

&nbsp;|AB| = |AM| + |BM| = {{{sqrt(abs(PM)^2-abs(PA)^2)}}} + {{{sqrt(abs(QM)^2-abs(QB)^2)}}} = {{{sqrt(((d*r[1])/(r[1] + r[2]))^2 - r[1]^2)}}} + {{{sqrt(((d*r[2])/(r[1] + r[2]))^2 - r[2]^2)}}}.

The problem is solved.

<H3>Example 1</H3>Find the length of the common interior tangent segment to two given circles in a plane, &nbsp;if their radii are &nbsp;3 cm&nbsp; and &nbsp;1.8 cm&nbsp; and the distance between their centers

is &nbsp;8 cm.

Solution

Use the &nbsp;Figure 2&nbsp; and the formulas derived in the solution of the &nbsp;Problem 2&nbsp; above.

According to this formula, &nbsp;the length of the segment &nbsp;AM&nbsp; is equal to

{{{sqrt(((d*r[1])/(r[1] + r[2]))^2 - r[1]^2)}}} = {{{sqrt(((8*3)/(3 + 1.8))^2 - 3^2)}}} = {{{sqrt((24/4.8)^2 - 3^2)}}} = {{{sqrt(5^2 - 3 ^2)}}} = {{{sqrt(25 - 9)}}} = {{{sqrt(16)}}} = 4 cm;

the length of the segment &nbsp;BM&nbsp; is equal to

{{{sqrt(((d*r[2])/(r[1] + r[2]))^2 - r[1]^2)}}} = {{{sqrt(((8*1.8)/(3 + 1.8))^2 - 1.8^2)}}} = {{{sqrt((14.4/4.8)^2 - 1.8^2)}}} = {{{sqrt(3^2 - 1.8^2)}}} = {{{sqrt(9 - 3.24)}}} = {{{sqrt(5.76)}}} = 2.4 cm.

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

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

My other lessons on circles in this site, &nbsp;in the logical order, &nbsp;are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/A-circle-its-chords-tangent-and-secant-lines-the-major-definitions.lesson>A circle, its chords, tangent and secant lines - the major definitions</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-longer-is-the-chord-the-larger-its-central-angle-is.lesson>The longer is the chord the larger its central angle is</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-chords-in-a-circle-and-the-radii-perpendicular-to-the-chords.lesson>The chords of a circle and the radii perpendicular to the chords</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/A-tangent-line-to-a-circle-is-perpendicular-to-the-radius-drawn-to-the-tangent-point.lesson>A tangent line to a circle is perpendicular to the radius drawn to the tangent point</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/An-inscribed-angle.lesson>An inscribed angle in a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/Two-parallel-secants-to-a-circle-cut-off-congruent-arcs.lesson>Two parallel secants to a circle cut off congruent arcs</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-angle-between-two-chords-intersecting-inside-a-circle.lesson>The angle between two chords intersecting inside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-angle-between-two-secants-intersecting-outside-a-circle.lesson>The angle between two secants intersecting outside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-angle-between-a-chord-and-a-tangent-line-to-a-circle.lesson>The angle between a chord and a tangent line to a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/Tangent-segments-to-a-circle-from-a-point-outside-the-circle.lesson>Tangent segments to a circle from a point outside the circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-converse-theorem-on-inscribed-angles.lesson>The converse theorem on inscribed angles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-parts-of-chords-intersecting-inside-a-circle.lesson>The parts of chords that intersect inside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/The-metric-relations-for-secants-intersecting-outside-a-circle.lesson>Metric relations for secants intersecting outside a circle</A> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Circles/Metric-relations-for-a-tangent-and-a-secant-lines-released-from-a-point-outside-a-circle.lesson>Metric relations for a tangent and a secant lines released from a point outside a circle</A>

under the topic &nbsp;Circles and their properties&nbsp; of the section &nbsp;Geometry, &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-bisect-an-arc-in-a-circle-using-a-compass-and-a-ruler.lesson>HOW TO bisect an arc in a circle using a compass and a ruler</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-find-the-center-of-a-circle-given-by-two-chords.lesson>HOW TO find the center of a circle given by two chords</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-a-radius-and-a-tangent-line-to-a-circle.lesson>Solved problems on a radius and a tangent line to a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-inscribed-angles.lesson>Solved problems on inscribed angles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/The-property-of-the-angles-of-a-quadrilateral-inscribed-in-a-circle.lesson>A property of the angles of a quadrilateral inscribed in a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Isosceles-trapezoid-can-be-inscribed-in-a-circle.lesson>An isosceles trapezoid can be inscribed in a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-tangent-line-to-a-circle.lesson>HOW TO construct a tangent line to a circle at a given point on the circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-tangent-line-to-a-circle-through-a-given-point-outside-the-circle.lesson>HOW TO construct a tangent line to a circle through a given point outside the circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/HOW-TO-construct-a-common-exterior-tangent-line-to-two-circles.lesson>HOW TO construct a common exterior tangent line to two circles</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-chords-that-intersect-within-a-circle.lesson>Solved problems on chords that intersect within a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-secants-that-intersect-outside-a-circle.lesson>Solved problems on secants that intersect outside a circle</A>,

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-a-tangent-and-a-secant-lines-released-from-a-point-outside-a-circle.lesson>Solved problems on a tangent and a secant lines released from a point outside a circle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/geometry/The-radius-of-a-circle-inscribed-into-a-right-angled-triangle.lesson>The radius of a circle inscribed into a right angled triangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-tangent-lines-released-from-a-point-outside-a-circle.lesson>Solved problems on tangent lines released from a point outside a circle</A>

under the current topic.

The overview of lessons on Properties of Circles is in this file &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Circles/PROPERTIES-OF-CIRCLES-THEIR-CHORDS-SECANTS-AND-TANGENTS.lesson>PROPERTIES OF CIRCLES, THEIR CHORDS, SECANTS AND TANGENTS</A>.
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 &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.