Apollonian cones have circular bases. The diameters of the base and the two cone generators that connect the ends of the diameter to the Apex form what Apollonius called an axial triangle.
Three cevians are associated with each
axial triangle. The median which is the line from the apex to the
center of the base, the apex angle bisector and the symmedian. The
symmedian is the reflection of the median in the the apex angle
bisector. Consequently the apex bisector also bisects the angle
between the median and the symmedian making them antiparallel with respect
to the apex bisector. i.e., the median and the symmedian make equal
angles in opposite direction with respect to the bisector.
When the apex of the cone is directly over the center of the base the axial triangles are isosceles and perpendicular to the base. The three cevians are collinear and lie on the intersection of the axial triangles. Such cones are called right circular cones.
If the apex is not over the center of the base it will instead be directly over a diameter of the base or an extension of a diameter. The axial triangle whose base is this diameter will be the only triangle that is perpendicular to the base and it will be referred to as the "upright" triangle. The triangle having as its base the diameter perpendicular to the base of the upright triangle will be only isosceles triangle.
The median of all the triangles will
be collinear and lie on the intersection of all the triangles. This
common median - line from the apex to the center of the circular base -
Apollonius called the cone axis and it is common to all axial
triangles. These cones are known as oblique or scalene cones.
The apex bisector that lies on the
upright triangle is in some texts is called the central axis because it is
the cone's axis of rotational symmetry. It is perpendicular to the
plane of the cone cross section and passes through the center of it.
Each axial triangle has an apex bisector but only the one that lies on the
upright triangle is of unique significance. The annotation in the
accompanying model relates to the cevians on the upright triangle.
Similarly, the symmedian is the reflection of the cone axis in the symmetry axis and it passes through the centers of the subcontrary circles which are reflections of the base circles in the cross section plane. The cone axis and the symmedian are isogonals.
Subcontrary and antiparallel are used
interchangeably and, as applied to cone sections, refer to sections that
are circular but not parallel. Such sections do not exist in
right circular cones but do in oblique ones.
The circumcircle of the axial triangle lies in the plane of the triangle and has a diameter that is perpendicular to the base of triangle at its center. The subcontrary is tangent to the circumcircle at the Apex making it perpendicular to the radius of the circumcircle drawn to the Apex.
The angles formed by the sides of the triangle and the line from the apex to the point where the perpendicular to the base circumcircle diameter intersects the circle below the base intercept equal arcs making the included angles equal. The apex bisector is collinear with this line. The angle formed by this line and a line from the apex to the top of the circumcircle will be inscribed in a semicircle and hence is a right angle.
On the upright triangle the apex
bisector is the cone symmetry axis. Thus the line from the apex to the top
of the circumcircle is perpendicular to the symmetry axis and parallel the
the cone cross section plane which bisects the angle between the base and
the subcontrary planes.
The model allows observation of the changes in the various entities that occur as the cone parameters change.