Apollonian Sphere and Properties of Stereographic Projection around the Lemoine Point

The definition and some of the properties of the Apollonian circle in the plane find their analogies in the Euclidean three-dimensional space. Thus, we manage to introduce a new concept in solid geometry that we call an “Apollonian sphere”. It appears that the Apollonian sphere not only possesses classical properties similar to the Apollonian circle such as orthogonality and coaxiality, but also analogies of its lesser-known connection with the Lemoine point and the circumcenter. We also discover two notable properties of stereographic projection that we prove with an Apollonian sphere. They include collinearity of the projection point with the Lemoine points of the projection and the projected triangles or with the centers of their Apollonian circles. Moreover, we connect the newly introduced concepts and the rich configurations they generate with Olympiad geometry.