TwelvePivots

1. Pivots

Given two triangles ABC and A'B'C' one pivot of inscription of A'B'C' in ABC is a point P, such that its pedal triangle with respect to ABC is similar to A'B'C'.

The basic properties of pivots are examined in the file Pivot.html . Here, fixing the angles of the triangles I examine the set of all pivots of A'B'C' relative to ABC. From corresponding properties of pedals (see references below) we know that the pedal of a point is similar to the pedals of the inverses of this point with respect to the Apollonian circles and also with respect to the circumcircle. These four inversions generate a finite group of 12 elements and the pivots of a definite couple of triangles builds an orbit of this group.

2. The group prototype

The group prototype G₀, to which the general group G we are interested in is isomorphic, is the one generated by two reflections {R₁, R₂} on two lines respectively {e₁, e₂} making an angle of 60 degrees, and one inversion F with respect to a circle (c) centered at the intersection of these two lines as in the following figure.

Group G₀ has 12 elements which are the three generators {R₁, R₂, F} the identity and the following eight:
(i) The reflection R₃ = R₂R₁R₂ on the line e₃ resulting by reflecting e₁ on e₂,
(ii) the two rotations V₁ = R₂*R₁, V₂ = V₁²,
(iii) and the six maps {F, FR₁, FR₂, FR₃, FV₁, FV₂}.

This group has the sector AOB between lines {e₁, e₂} and the circle (c) as fundamental domain i.e. every orbit G₀x = {g₁x, ..., g₁₂x} of the group, for x arbitrary point in the plane, meets this domain in exactly one point. Points inside the domain have orbits of exactly 12 points whereas points on the boundary have orbits with fewer than 12 points. More precisely points on the open arcs {OA, OB, AB} have orbits with 6 points, points {A,B} have orbits of 3 points and point O has an orbit of two points, which are O and the point at infinity I.

2. The Lemoine group of a triangle

Given a triangle ABC there is a group G of twelve elements generated by the inversions on its Apollonian circles and also on its circumcircle. This group is isomorphic to the group G₀ of the previous section through conjugation with a Moebius transformation H. H is defined by the requirement to map the vertices of an equilateral A₀B₀C₀ correspondingly to the vertices of the triangle ABC.

H maps the symmetry axes of the equilateral to the Apollonian circles and the circumcircle of it to the circumcircle of ABC. Point O is maps via H to the first isodynamic point J of triangle ABC. Every element F of the group G is conjugate by H to some element F₀ of the group G₀ i.e. F = HF₀H^-1. The orbits Gy of a point y under G are images H(Gx) of orbits of the corresponding point x which maps to y(= H(x)). These facts are proved in the file Moebius.html . There it is also proved that the circles centered at O map via H to the circle bundle generated by the circumcircle and the Brocard circle of ABC, hence the orbits of y by G are by six on two circles of this bundle which are inverse with respect to the circumcircle.

3. The 12 pivots

By the analysis made here and in Pivots.html the pivots of inscription of a triangle A'B'C' into ABC are orbits Gy of the group G. Hence they contain 12 points if y is in the fundamental domain of the group which is the image of the fundamental domain of G₀ under H. Thus there is in general a set of 12 pivots of inscription of A'B'C' into ABC. But if y is on the circumcircle arc CC' then the pedal triangle is degenerate (a Simson line), whereas if y is on the open arcs {JC, JC'} there are six pivots. This case is characteristic for pivots having pedals isosceli. Last case is when y coincides with J. In this case there are two pivots coinciding with the two isodynamic points. The corresponding pedal triangles are equilateral. A picture of the six pivots lying inside the circumcircle of the triangle can be viewed in Pivots.html .

1. Pivots

2. The group prototype

2. The Lemoine group of a triangle

3. The 12 pivots

See Also