The main properties of Moebius transformations are:

[1] They build a group, the composition corresponding to the product of matrices.

[2] They are

[1] They preserve the cross ratio of four complex numbers.

[3] They map the set of circles and lines to itself.

[5] They are completely determined by three points in general position and its images.

[1] F maps the circumcircle of Á

[2] F maps the symmetry axes of A

[3] The center of A

[4] The bundle of lines through the center of A

The proofs are easy consequences of the listed properties of Moebius transformations.

[1]: Follows from the preservation of circles.

[2]: The image of the symmetry axis A

[3]: Is a consequence of [2], since the isodynamic points are the common points of the Apollonian circles.

[4]: Is a consequence of [3] and the conformality of F. Bundle (II) is the orthogonal one to (I) as this happens by their pre-images through F.

In the case (e) is a line consider first the point at infinity I and its image J=F(I) by F. Since every line passes through I, its image will pass through J. Thus, circle k=F(e) and the image of a line k

Since F is conformal k

If (e) is a circle the argument is the same. One has only to consider points {z, z', z

In connection with the settings of the previous theorem

[1] The inversions with respect to the Apollonian circles of ABC are conjugate via F to the reflexions on the symmetry axes of A

[2] The inversion w.r. to the circumcircle of ABC is conjugate via F to the inversion w.r. to the circumcircle of A

[1] The pre-images {K

[2] The intersection points {L

These are consequences of the fact that the Brocard circle, having diameter QK and the Lemoine axis belong to the bundle (II) which is the image via F of the circles centered at O.

[1] The Brocard points {W, W'} of triangle ABC have pre-images under F the other two vertices of the equilateral triangle inscribed in the circle with diameter K

[2] The Brocard angle of triangle ABC is equal to the angle between line (e) and the circle (f

The first claim follows from the fact that the circle of bundle (I) passing through {J,J'} and the Brocard point W intersects the Brocard axis by an angle of 120 degrees. Hence its preimage, which is line OW

(i) The equilateral A

(ii) The line (e) through the center of the equilateral,

(iii) The point I

Fixing the vertices of the equilateral, it seems that we can parameterize all triangles of the plane by means of pairs (e, I

There is though a fine point that should be noticed. Assume that the two triangles A

is a Moebius transformation, which maps one triangle onto the other. G respects also the isodynamic points, since their pre-images under the F

The above image illustrates the case. Lines {b

There are though cases for which G(b

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