This
investigation begins with the following problem.
Given two circles and a point on one of the circles. Construct a circle tangent to the two circles with one point of tangency being the designated point.
The problem asks us to construct a circle that is tangent to 2 other circles. Tangent circles are circles that connect at a single point. We will attempt to construct a circle that is tangent to 2 other circles. The first thing we will do is draw the 2 circles. We will place any point on the bigger circle and call this point C. Next we construct a line that passes through both point A which is the center of the larger cirlce and point C. After constructing this line we will construct a circle with the radius of the smaller cirlce and centered at point C.
The Orange circle that is centered at C will be used to construct the circle that is tangent to both the circle centered at A and the circle centered at B. Now we will construct the point at the intersection of the circle centered at C and the line passing through AC. We will call this point D. Now we will construct a segment from Point D to point B. Then construct a perpendicular bisector of segment BD. The intersection (which we will call point E) of perpendicular bisector and segment AC will be the center of the circle that is tangent to both the circle centered at point A and the cirlce centered at point B.
Here is a cleaned up version of the circle that is tangent to both circles. You can also maniuplate the script tool in GPS.
Lets also consider the case where the smaller circle is inside of the tangent circle. Again we will construct a circle centered at Point A and a circle centered at point B. This time Point C will be placed anywhere on Circle B. Now we will construct a point that crosses through point B and Point C. After constructing the line we will construct a circle centered at C with the radius of Circle A.
The orange circle is centered at C and will be used to construct the circle that is tangent to both the circle centered at A and the circle centered at B. Now we will construct the point at the intersection of the circle centered at C and the line passing through BC. We will call this point D. Now we will construct a segment from Point D to point A. Then construct a perpendicular bisector of segment AD. The intersection (which we will call point E) of perpendicular bisector and segment BC will be the center of the circle that is tangent to both the circle centered at point A and the cirlce centered at point B. The circle centered at point B will be inside of the tangent circle.
Here is a cleaned up version of the tangent circle. You can also manipulate the script tool in GPS.
We will investigate and discuss the loci of the of the centers of the tangent circles for three different cases.
i. When one circle lies completely inside the other.
ii. When the 2 circles overlap.
iii. When the 2 circles are disjoint.
i. When one circle lies completely inside the other.
For thie first example we have the circle centered at Point B completely inside of the circle centered at point A. We will trace point E which is our tangent circle. as we rotate point C we can see that point E creates an ellipse. The foci of the ellipse is points A and B which are the centers of the original circles.
ii. When the 2 circles overlap.
For the second example we have the 2 circles overlapping. We will rotate point C around The circle centered at point A. We will again trace point E which is the center of our tangent circle. As we rotate point C we can see that point E again creates an ellipse. The foci of the ellipse is again points A and B which are the centers of the original circles.
iii. When the 2 circles are disjoint.
For the third example we have 2 disjoint circles. We will animate point C around the circle centered at point A. We will again trace point E which is the center of the tangent circle. As we rotate point C we can see tht point E creates a Hyperbola. The foci of the hyperbola is points A and B.