Soddy Circles

 

Given three noncollinear points A, B and C, we can construct three circles whose centers are situated at A, B and C and pair wise tangent to one another. Then there are exactly two circles that are tangent to all the three circles. Such circles are called soddy circles. The one lies inside of the three circles is called inner soddy circle. The other lies outside is called outer soddy circle. What's more, connect the points of tangency of these circles with lines, we will find two concurrent points.(Pt1,Pt2)

It's easy to construct the first three tangent circles. As to their soddy circles, we need the following property of inversion to find them:

The inversion of any line L, not through the center of inversion circle O, is a circle through O, and that the diameter through O of this circle is perpendicular to L. Conversely the inversion of any circle C, through O, is a line perpendicular to the diameter through O.

 

 

 

Construction:

1 Given three points A, B and C. They form a triangle.
 
2 Use angle bisectors to find the inner center of the triangle.
 
3 Construct the lines which perpendicular to AB, BC, AC. Label the intersection points A', B', C'.
 
4 Draw the following three circles: center at A and through B', center at B and through C', center at C through A'.
 
5

Now open a new file to define a macro and named it "ivclcl", which means inversing a circle to a circle.

Construct two circles. One is the inversion circle, which we label as "iv". The other one, circle Q, is the circle we want to inverse. Connect the centers with a line. Label the intersections of circle Q and the line as point S and point T.

 
6

Construct the point S' and the point T',the inverse of S and T. Then the inverse of the circle Q is the circle which has a diameter with endpoints S' and T'.

Initial objects: circle Q and circle iv.

Final objects: the red circle.

 
7 Open another new file to define the macro and named it "ivclln", which means inversing a circle to a line. Construct two circles. As above, one is inversion circle, the other circle Q is the circle we want to inverse and passes the center of circle iv.
 
8 Connect points iv and Q with a line. Label the intersection of the line and circle Q with S.
 
9 Find the point S', the inversion of S. Construct the line perpendicular to the line ivS' and through S'.
 
10 Back to the first file to continue finding soddy circles. Construct circle B', whose center is B' with any radius, not through B. And use the macro "ivclln" to fine the inversions of circle A and circle C. Find the inversion of circle B with the macro "ivclcl". We will get two parallel lines A*,C*and a circle B* which tangent to A* and C*.
 
11 Construct circles S1 and S2, which are tangent to A*, B* and C*.
 
12 Then the inversion of S1 is the inner soddy circle, the inversion of S2 is the outer soddy circle.
 
13  
 
14 The concurrent of the inner soddy circle.
 
15 The concurrent of outer soddy circle.
 

Source from: http://mathworld.wolfram.com/SoddyCircles.html