Question 1068804
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Two circles,centre A and B touch one another at C.Through C a straight line PCQ is drawn cutting the circles at P and Q.Prove that AP || BQ.
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<pre>
The plot is shown in the Figure.


{{{drawing( 300, 300,  -7.0, 11.0, -9.0, 9.0, 
            circle( -3.0, 0.0, 3.0), 
            circle( -3.0, 0.0, 0.2), 

            circle(  5.0, 0.0, 5.0), 
            circle(  5.0, 0.0, 0.2), 

            line( -6.5, 0.0, 10.5, 0.0),

       blue(line( 0.0, -6.0, 0.0, 6.0)),
            locate( 0.2, -5.0, E),
            locate( 0.2,  5.0, F),

            line( 0.0, 0.0,        8.0,      4.0),
            line( 0.0, 0.0, -3-0.6*3.0, -0.6*4.0),

            locate(-3.25,- 0.1, A),
            locate( 4.75, -0.1, B),
            locate( 0.3,  -0.1, C),

            locate( -4.8-0.2,  -2.6, P),
            locate(   8.1,      4.7, Q),

        red(line( 5.0, 0.0,        8.0,      4.0)),
        red(line(-3.0, 0.0, -3-0.6*3.0, -0.6*4.0))
)}}} 


Draw EF, the straight line perpendicular to AC at the point C.
Then EF is a tangent line to the circle A.
At the same time, DE is the perpendicular to BC at the point C and is a tangent line to the circle B.
So, the line BCA is a straight line.

The angles ACP and BCQ are congruent as they are vertical angles.    ( See the lesson (*) )

The angles ECP and FCQ are congruent due to the same reason.         ( See the lesson (*) )

Therefore, the minor arcs CP and CQ have the same measures.          ( See the lesson (**) )

It implies that the central angles PAC and QBC are congruent.

Thus the triangles ACP and BCQ have two pairs of congruent angles:
(ACP and BCQ, as well as CAP and CBQ).

Hence, the angles APC and BQC are congruent as the pair of the third angles of triangles.

But these angles are <U>alternate interior angles</U>.                      ( See the lesson (***) )

It implies that the straight line AP is parallel to BQ:  AP || BQ.   ( See the lesson (***) )

QED.
</pre>Solved.


<pre>
The referenced lessons are 

&nbsp;&nbsp;&nbsp;&nbsp;(*) &nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Angles/Vertical-Angles-.lesson>Vertical angles</A>
&nbsp;&nbsp;&nbsp;&nbsp;(**) &nbsp;&nbsp;<A HREF=https://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;(***) &nbsp;<A HREF=https://www.algebra.com/algebra/homework/Angles/Parallel-lines.lesson>Parallel lines</A>
</pre>

Also, &nbsp;you have this free of charge online textbook on Geometry

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A> 

in this site.


The referred lessons are the part if this textbook under the corresponding topics.