In fact, the right-angled triangles {PA'A'', PB'B'', PC'C''} are all similar. This implies the similarity of triangles {PA'B', PA''B''} and also of triangles {PB'C', PB''C''}. From these similarities follows also the claimed similarity.

The first possibility gives 6 cases and the second doubles them to 12. Thus, there are at most twelve different locations for P relative to ABC and consequently also relative to A'B'C'. In the previous figure we have a case in which the orientation of A'B'C' is the inverse of the orientation of ABC (see Inverse_Pedals.html ).

Consequently (Corollary-2 above) also at most twelve possible positions of P relative to ABC.

Next figure shows the positions of these twelve points relative to A'B'C' (the inscribed one) in a case of two triangles for which the twelve positions are all different. The positions relative to triangle ABC present a different and more interesting structure, which is studied in the file TwelvePivots.html .

In fact, given the position of P relative to A'B'C', draw the circumcircles of triangles {PA'B', PB'C', PC'A'}. Take then some point A on the first circle and join it to some vertex of A'B'C' lying on the same circle. Extend the previous line to intersect a second time the adjacent circle at B and repeat the procedure from B to find C. An angle chasing argument shows that {A, C, B'} are collinear and the triangle thus constructed has the correct relations of angles to the angles {PA'B', PB'C', PC'A'}.

The construction of a triangle with this prescriptions (similar to A'B'C' and inscribed in this way in ABC) can be carried out in the following way.

Select an arbitrary point A

Let C

TwelvePivots.html

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