## In this investigation we will be exploring tangent circles given two fixed circles, A and B. Given two circles, there are two cases for the construction of a third circle that is tangent to both of them. We will look at these two cases separately.

## Case 1: Assume circle B is within circle A, and we want to construct a tangent circle that is within circle A and between circle B and the exterior of circle A.

Note: See the picture that follows for a visual of this construction.## Step 1: Given circle A and circle B, mark any point on the circumference of circle A (call it X).

## Step 2: Construct circle X with center X and the same radius as circle B.

## Step 3: Construct a line through points X and A. Mark the point where this line exits circle X and call it Y.

## Step 4: Construct the line segment connecting points Y and B, and construct its midpoint (call it M).

## Step 5: Construct a line perpendicular to YB through M. Mark the intersection of this line with XA and call it D.

## Step 6: Construct circle D with center D and radius DX.

## Circle D (the green circle) is tangent to circle A and circle B (the blue circles).

For a GSP file containing a script tool for this construction, click here.## Case 2: Assume circle B is still within circle A, but now we want the tangent circle to contain circle B.

Note: See the picture that follows for a visual of this construction.## Step 1: Given circle A and circle B, mark any point on the circumference of circle A (call it X).

## Step 2: Construct circle X with center X and the same radius as circle B.

## Step 3: Construct a line through points X and A. Mark the point where this line intersects the circumference of circle X on the interior of circle A; call this point T.

## Step 4: Construct the line segment connecting points T and B, and construct its midpoint (call it M).

## Step 5: Construct a line perpendicular to segment TB through point M. Mark the intersection of this line with XA and call it E.

## Step 6: Construct circle E with center E and radius EX.

## Circle E (the purple one) is tangent to circle A and circle B (the blue ones).

For a GSP file containing a script tool for this construction, click here.

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