New theorem : Use of an inversion

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

In this Paragraph, we suppose that points $A_i$ are on a circle $\Gamma$ of center O and radius R .i will stanf for the inversion of center O and ratio $R^2$.

Proposition 2.1 : We have T o i = i

Demonstration : Let’s have a point M of plane E and have N = i (M). The circle $\Gamma$ being invariant through i, proposition 1.4 gives us :

$N{A_i} = \frac{{R^2} M{A_i}}{OM \times O{A_i}} = \frac{R}{OM} \times MA_i$ .

The coefficients NA_i being proportional to coefficients MA_i, we deduct that the barycenter of points A_i,MA_i) and (A_i,NA_i) are identical, so we get T(M)=T(N)=(Toi)(M).

Corollary 2.2 : The image of plane E through T is equal to the image through T of the inside of the cercle of center O and radius R (circle included).

Indeed, if a point M is not inside the circle, it’s image through i is.

Continuation of the demonstration

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