Tangents to Circles
by
Susan Sexton
For this
exploration I will construct the common tangents to two given circles. Lets see the different configurations
for two different circles.
Two distinct non-intersecting circles:
Two distinct circles that share
one common point:
Two circles that share two common points:
Two Distinct Non-Intersecting Circles
Let me first
explore two distinct non-intersecting circles. There are two possible tangent lines, an external tangent
line and an internal tangent line.
Externally
Tangent Lines
Given two
circles, circle A (whose center is at point A) and circle B (whose center is at
point B), construct the line passing through A and B. Construct radius AC of circle A. Construct a line parallel to radius AC through B that
intersects circle B at point D.
Construct the
line passing through C and D which intersects line AB at point E.
Construct a
circle of diameter EB by finding the midpoint of EB, point M, and using M as
the center with radius MB.
Construct the
line through E and the two points (points R and S) of intersection of circles M
and B.
Line ER and line
ES will be the external tangent lines to the given circles, circle A and circle
B.
Going Back -
This construction
can be traced back to the construction of the tangent lines to any given circle from a point
exterior to the circle. Here we
have circle C and point P.
Construct segment
PC and find the midpoint of PC, point M.
Next construct circle M with radius MC.
The two points of
intersection between the two circles will be the points of tangency (points R
and S) from P to circle C.
If you have two distinct circles then find the point of intersection of the two tangent lines (which will be exterior to both circles) to the bigger circle. But make sure the line created to construct circle M goes through the center of both circles. Unless the circles are congruent then the external tangents
will intersect at a point outside. If the circles are congruent then the tangent lines will be parallel but the construction will still work.
Internally
Tangent Lines
Next I will
construct the internally tangent lines to two given circles. Given circle A (whose center is at
point A) and circle B (whose center is at point B), construct segment AB. Construct a line perpendicular to AB
through A and another line perpendicular to AB through B.
The newly
constructed line through A will intersect circle A at points C and D. The newly constructed line through B
will intersect circle B through points E and F.
Construct segment
CF that intersects AB at point G.
Construct a
circle whose diameter is AG. This
can be done by finding the midpoint of AG, point M, and using radius MG. This circle will intersect circle A at
points R and S.
The line through
G and R and the line through G and S will be internally tangent to both
circles.
Notice how the
construction of internally tangent lines requires a new circle, circle M, much
like the construction of externally tangent lines. This construction can be traced back to finding the tangent
lines to a circle from an external point.
In this case, it is point G that is external to both circles.
Two
Distinct Circles That Share a Common Point
Given two
distinct circles A and B that share one common point C.
Whether one
circle lies in the interior of the other or not,
This is because
the line tangent to both circles will be at point C. Since the tangent line is perpendicular to a circle at its
point of tangency then B will lie on the line perpendicular to the line through
C. Similarly, A will also lie on the line perpendicular to the line through C. Since there can only be one line
perpendicular to a given line through a point on the given line then A and B
must be collinear. So by
constructing the line through A and B then the line will also go through
C.
Now just
construct the line perpendicular to line AB through C in either case.
Two
Distinct Circles That Share Two Common Points
The construction
of the tangent lines to two distinct circles that share two common points will
have to be external tangents.
Otherwise the lines will intersect the circles in more than one point
and will no longer be tangent lines.
Therefore we can use the same construction as if the two circles share
no common points as illustrated below.
There are some really foundational ideas that are coming out of this exploration. These ideas include:
Can two distinct circles intersect in more than 2 points?
Is a perpendicular line to a given line through a point on the given line really unique?
Constructing circle M in
both cases above was really key in obtaining the tangent lines. What is unique about the circle M?
By exploring
these ideas, some key geometric ideas can be explored to solve even more
difficult problems.
Here are some GSP
sketches and scripts that can help in constructing tangent lines.
Tangent Lines to
Two Circles That Share a Common Point