Assignment 7: Tangent Circles
By Nikhat Parveen, UGA
Given a arbitrary line L and a circle(C1) with center K and an arbitrary point P on the circle, We would like to construct two circles tangent to the given circle at P and tangent to the line L.
Let's begin by constructing a perpendicular line to the given line L through center K. Call Q. The line Q intersects the given circle C1 in two points M and N. Draw lines through P and M and P and N. These two lines intersect line L in two points A and B. Construct two perpendicular lines to line L through point A and point B. Call them S and T. Draw another line through point P and center K of circle C1. This line intersects the perpendicular lines S and T in two points O and G.
Now O and G forms the center of our two desired circles. Construct two circles with center O and center G. These two circles will pass through the point P as desired.
Click here for GSP file.
Some interesting explorations!!!!
Let's animate the point P which is also the point of contact for the two circles we found. And let's trace the center O and G of the two circles (purple and pink). What do you think might happen? let's explore.....
Click HERE for the animation file in GSP
Well, what have we got here, two parabolas. Why do you think the centers we traced turned out to be a parabola? What can we conjecture from this?
If we consider line L as the directrix and point P as focus then the locus of points equidistant from the line L (directrix) and the fixed point P (called focus) is a parabola as per the definition.
Now if we construct a line segment joining PA and PB and take their midpoint (call a and b)and construct a perpendicular line passing through the midpoints and perpendicular to the segment PA and PB, we get the line tangents (call T1 and T2)to the parabola's in RED and BROWN.
Let's trace the tangent lines T1 and T2..........
Click here for the GSP file
If you look at the figure 2, line through PM and PN intersects the tangent circles in point J and H. Let's trace these points. What do you expect the locus of these points would like. I expect it to be a parabola again because we are tracing the locus of points equidistant from the line L (directrix) and the fixed point P(called focus)
Please click HERE for the animation file in GSP
Well, our expectation was right!! Notice that the locus of point H is a parabola tangent to the given circle at the bottom (point M) and the locus of the point J is the parabola that is also tangent to the given circle at the top (point N). If you also notice the line Q passing through the center K and perpendicular to line L is the axis of symmetry to all the parabolas we have drawn so far.
Conjecture: If two circles tangent to the given circle at an arbitrary fixed point P and to given arbitrary line L then the locus of their centers is a parabola.
Click HERE to explore some more hidden parabolas.
We can further explore this investigation by moving the given circle to different locations. For example I tried to place the given circle external to one of our tangent circles and found something very neat.
Click here for the GSP file
The locus of the point O (center of purple circle) and J (point on the purple circle) are the inverted parabolas whereas the locus of center G (pink circle) and point H (on pink circle) are the parabolas that are stretched as compare to the previous graphs. Also we can notice a section of hyperbola. Also notice that the line T1 and T2 are still tangent to the parabolas (brown and red).
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