The theorem concerning the three points on a conic A, B, and C, whose
osculating circles pass through a fourth point O on the conic, is due
to Steiner. From the properties of the harmonic polars of the points of
inflection on a nodal cubic we may infer many other theorems concerning
the points A, B, and C on a conic. Let the cubic be projected into
a circular cubic and then inverted from the node. Its points of
inflection A₁, B₁, C₁ invert into the points A, B, and C. The harmonic
polar of A₁ inverts into the common chord O P of the circles osculating
the conic at B and C; and similarly for the other harmonic polars.
The pencil O {A B P C} is harmonic. Any circle through A and O meets
the conic in S and T so that the pencil O {A S P T} is harmonic. The
two circles through O and tangent to the conic at S and T intersect on
O P. If two circles be drawn through O and A intersecting the conic
one in S and T and the other in U and V, the circles O S U and O T V
intersect on O P; so also the circles O S V and O T U. But one circle
can be drawn through O and A and tangent to the conic; its point of
contact is on O P. Let l, m, and n be three points on the conic on a
circle through O. Draw the circles O A l, O A m, and O A n intersecting
the conic again in l₁, m₁, n₁; l₁, m₁, n₁, are also on a circle through
O, and the circles through l, m, n and l₁, m₁, n₁ intersect on O P.