Jadonna Brewton

Fall 2000

Assignment #7

Problems # 1-8

Tangent Circles

Given two circles and a point on one of the circles, construct a circle that is tangent to the two circles with one point of tangency being the given point.

SMALLER CIRCLE INSIDE THE LARGER CIRCLE

Case 1: The given point E is on the larger circle. The smaller circle is external to the tangent circle (center at M).

Note: The locus of the center M of the tangent circle as point E travels around the large circle is an ellipse with the centers of the two circles (points A and C) as the foci.

 

 

                         
SCRIPT                                                 GSP SKETCH

Case 2: The given point E is on the larger circle. The smaller circle is internal to the tangent circle (center at I).

Note: The locus of the center I of the tangent circle as point E travels around the large circle is an ellipse with the centers of the two circles (points A and C) as the foci.


                                 
SCRIPT                                             GSP SKETCH

Case 3: The given point E is on the smaller circle. The smaller circle is external to the tangent circle (center at H).

Note: The locus of the center H of the tangent circle as point E travels around the small circle is an ellipse with the centers of the two circles (points B and D) as the foci.

 

                      
SCRIPT                                     GSP SKETCH

Case 4: The given point E is on the smaller circle. The smaller circle is internal to the tangent circle (center at H).

Note: The locus of the center H of the tangent circle as point E travels around the small circle is an ellipse with the centers of the two circles (points B and D) as the foci.




 



                           
SCRIPT                                     GSP SKETCH






INTERSECTING CIRCLES



Case 1:
The given point E is on the larger circle. Two tangent
circles are constructed. In the picture below, the small circle
is external to the red
tangent circle (center at J) and
internal
to the green tangent circle
(center at L).



Note:
As point E travels around the large circle,




  
(1) the locus
  of the center J of the red tangent circle is an ellipse
  with the centers of the two circles (points A and C)
  as the foci.

  
(2) the locus
  of the center L of the green tangent circle is a hyperbola with
  the centers of the two circles (points A and C)
  as the foci.



            
SCRIPT                                     GSP SKETCH



 






Case 2:
The given point E is on the smaller circle. Two tangent
circles are constructed. In the picture below, the large circle
is external to the red
tangent circle (center at I) and
internal
to the green tangent circle
(center at K).



Note:
As point E travels around the large circle,




  
(1) the locus
  of the center I of the red tangent circle is an ellipse
  with the centers of the two circles (points A and C)
  as the foci.

  
(2) the locus
  of the center K of the green tangent circle is a hyperbola with
  the centers of the two circles (points A and C)
  as the foci.



      
SCRIPT                                          GSP SKETCH




DISJOINT CIRCLES



Case 1:
The given point E is on the larger circle. Two tangent
circles are constructed. In the picture below, the small circle
is internal to the red
tangent circle (center at J) and
external
to the green tangent circle
(center at L).



Note:
As point E travels around the large circle,




  
(1) the locus
  of the center J of the red tangent circle is a hyperbola
  with the centers of the two circles (points A and C)
  as the foci.

  
(2) the locus
  of the center L of the green tangent circle is a hyperbola with
  the centers of the two circles (points A and C)
  as the foci.



        
SCRIPT                                      GSP SKETCH






Case 2:
The given point E is on the smaller circle. Two tangent
circles are constructed. In the picture below, the large circle
is internal to the red
tangent circle (center at I) and
external
to the green tangent circle
(center at K).



Note:
As point E travels around the large circle,




  
(1) the locus
  of the center I of the red tangent circle is a hyperbola
  with the centers of the two circles (points A and C)
  as the foci.

  
(2) the locus
  of the center K of the green tangent circle is a hyperbola with
  the centers of the two circles (points A and C)
  as the foci.



       
SCRIPT                                      GSP SKETCH






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