Collineation of period three and triply perspective triangles (Q750887)

The authors give synthetic proofs of some theorems in classical projective geometry over a field. As they remark themselves, most of their results were previously proved algebraically by \textit{S. N. Collings}, Math. Gaz. 53, 265-269 (1969; Zbl 0181.238). The essential theorems are the following ones. Let f be a projective collineation of order 3, and let \((A,A^ f,A^{f^ 2})\), \((B,B^ f,B^{f^ 2})\) be two triangles in general position. Then these triangles are triply perspective, and f exchanges the three centres of perspectivity cyclically. If two triangles (A,B,C) and \((A',B',C')\) are doubly perspective, then they are triply perspective, and there is a collineation of order 3 such that \(A^ f=B\), \(B^ f=C\), \(A^{'f}=B'\), \(B^{'f}=C'\) (up to change of notation).