Consider the homography F, which recycles the triangle vertices C- >B- >A- >C and simultaneously preserves the circumcircle g of the triangle (F(g)=g). The orbit of E is F(E) = I and F(I) = L. The orbit of the middle X of LI is F(X) = O and F(O) = P. The points O and P are the Brocard points of the triangle. The isosceli triangles EPO, XLO, XPI are similar. Line e is mapped via F to the line-at-infinity. e is also orthogonal to EL. Analogously the symmetric e' of e w.r. to EX is mapped via the invers G of F to the line at infinity. L is the pole of e w.r. to the circle g. R is the intersection point of the symmedians. The Brocard's ellipse b remains invariant w.r. to F. The same happens with every member of the family of conics generated by g and b. In the picture there is a point D and its image K = F(D) by F.

An other recycler of the triangle is described in the file: Recycler.html .

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