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This article demonstrates a new theorem I discovered twelve years ago while testing MathGraph32. This theorem is a direct consequence of Ptolemy’s theorem.

0. Introduction

I. The preliminary results we will use in this article

II. Case of n cocyclic point : Use of an inversion

III. The case of two points

IV) The case of three cocyclic points. Image through *T* of the circumscribed circle to these points

V) The case of points A_{i} cocyclic. Image through *T* of the circumscribed circle of these points

VI. A few properties of triangle A’B’C’ in the case of three points

VII. Many questions are still open for research

**1. Definition of a transformation T.**

E is an euclidien plane.

**Definition of T (M) :** If *n* is an integer, $n \ge 2$, we define the function $T_{A_1A_2...A_n}$ de E dans E by : $T_{A_1A_2...A_n}(M)$ = bar {(A_{1}, MA_{1}) ; (A_{2}, MA_{2}) ; … ; (A_{n}, MA_{n})}. (bar standing for barycenter).

When there is no ambiguity, this function will simply named *T*.

**2. Presentation of this article.**

We will study first the case of two points, then three points, finally *n* points.

In the case of *n* points ($n \ge 4$) a lot of issues are still open for research which we will see at the end of this article.

We will identify E and the complex plane **C** and we will then name $a_1, a_2, …, a_n$ the complex affixes of $A_1, A_2, …, A_n$.

**3. How were these results discovered ?**

Of course I am not absolutely sure that it is a discover, but until now I never met anybody else who claimed this discovery before.

It is while testing my software MathGraph32, that I fell on a result that seemed very strange. I had had the idea to create a triangle ABC with it’s circumscribed circle. Then I had created a point M linked to this circle, measured lengthes MA, MB et MC and created the barycenter M’ of the balanced points (A, MA), (B, MB) et (C, MC). To my great surprise, I saw that the locus of points M’ when M was moving around the circle seemed to be a triangle. Then I discovered that this result was a consequence of Ptolemy’s theorem.

**4. Credits.**

I have to thank particularly Mahdi Abdeljaouad, professor in the university of Tunis, for some advices, particularly the idea of using transformation notations and mainly Daniel Perrin, professor in the university of Orsay (France), who completed the demonstration I had done, studying fully the transformation *T* in the case of two and three points and by using in a very sensible way the inversion.

**5. The article’s figures**.

They are all dynamic, animated by the MathGraph32’s JavaScript library. You can capture the moveable points.

It seems that the results obtained for three points are still valid for n points, meaning that if A_{1}A_{2}…A_{n} is a convex polygon inscribed in a circle $\Gamma$ of center O and radius R, the image of the plane through $T = {T_{{A_1}{A_2}...{A_n}}}$ is the inside of the polygon A’_{1}A’_{2}…A’_{n} where the points A’_{i} are the images of points A_{i} through T and that the points M’inside this polygon have only two antecedents through T which are the images of each other through the inversion i of center O and ratio R^{2}.Figures made with MathGraph32 seem to confirm this result, such as the figure beside where I created for five points the images through T of the circles C_{E,D,k}. |

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