Soddy circles revisited
This constructions is inspired by the construction given by Raymond in http://www.geogebra.org/for....
Given a triangle ABC draw the hyperbola with focus B and C through the point A. The point D is the intersection of the branch through the point A. With the circles with centres B and C through D, define the point E and F. The circle with center A through E is tangent to the two first circles. To find the centres of the inner and outer Soddy circles, draw, for example the hyperbola through the point B with focus A and C. The intersection G of the branches of the the hyperbolas through the points A and B is the centre of the inner Soddy circle and the intersection K of the branches not through the points A and B is the centre of the outer Soddy circle.
The construction is not new. In fact, the hyperbolas are called the Soddy hyperbolas of the triangle ABC.
Best regards, jtico