2.32Desargues’ Theorem. If two triangles are perspective from a point they are perspective from a line
Let two triangles PQR and P’Q’R’ be perspective from a point O. We see from Axiom 2.14 that their three pairs of corresponding sides meet, say in D,E,F. Consider the triangles PP’E and QQ’D. Since pairs of corresponding sides meet in the three collinear points R’, R, O, these triangles are perspective from a line, and therefore (by 2.31), perspective from a point, namely from the point PQ.P’Q’=F. That is, the three points E,D,F are collinear. The converse of the theorem can be deduced by applying 2.32 to the triangles PP’E and QQ’D.