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.

We can think three cases in the situation above(given circles have red colors) and in each case we can get two kinds(pink color and blue color) of tangent circle like the following.(Of couse, there are tangency cases between each cases)

Case1. One circle is included in another circle

Case2. One circle intersects with another circle in two different point

case3. One circle lies outside of another circle

How does the locus for the center of the tangent circle be drawn?

It's very interesting to investigate the situation above while shifting continuously from case1 to case3.

Let's consider the method to make the tangent circle at first.

Method 1 When the smaller circle is external to the tangent circle.

<<<< Pink Circle >>>>

(1) Draw the line passing through a center of the bigger circle

(2) Draw a circle with radius of the smaller circle at an intersection getting from (1)

(3) we can get the isosceles triangle above using the property of perpendicular bisector

(4) Our pink circle becomes tangent to two red circles

What is the locus of C?

OC + O'C = OC + CD = Sum of radiuses of two given circles = constant

Therefore the locus of C is ellipse

Invetigate and let's see the result figure

<<<< Blue Circle >>>>

(1) Draw the line passing through a center of the bigger circle

(2) Draw a circle with radius of the smaller circle at an intersection getting from (1)

(3) we can get the isosceles triangle above using the property of perpendicular bisector

(4) Our blue circle becomes tangent to two red circles

What is the locus of C?

OC + O'C = OC + CD = Difference between radiuses of two given circle = constant

Therefore the locus of C is ellipse

Invetigate and let's see the result figure

Method 2 When one circle intersects with another circle in two different point

We can get the isosceles triangle above as moving of the smalller circle from the method1 using the funtion of GSP.

Our pink and blue circle becomes tangent to two red circles

<<<< Pink Circle >>>>

What is the locus of C?

OC + O'C = OC + CD = Sum of radiuses of two given circles = constant

Therefore the locus of C is ellipse

<<<< Blue Circle >>>>

What is the locus of C?

O'C - OC = CD - OC = Difference between radiuses of two given circle = constant

Therefore the locus of C is hyperbola

Invetigate and let's see the result figure

Method 3 When one circle lies outside of another circle

We can get the isosceles triangle above as moving of the smalller circle from the method1 using the funtion of GSP.

Our pink and blue circle becomes tangent to two red circles

<<<< Pink Circle >>>>

What is the locus of C?

OC' - OC = CD - OC = Sum of radiuses of two given circles = constant

Therefore the locus of C is hyperbola

<<<< Blue Circle >>>>

What is the locus of C?

O'C - OC = CD - OC = Difference between radiuses of two given circle = constant

Therefore the locus of C is hyperbola

Invetigate and let's see the result figure