If two triangles are in perspective from a point (lines through corresponding vertices meet at one point), then they are in perspective from a line (corresponding sides meet on one line) — and vice versa.
Let △ABC and △A'B'C' be two triangles. If lines AA', BB', CC' are concurrent (meet at point O — the "center of perspectivity"), then the three intersection points of corresponding sides — AB∩A'B', BC∩B'C', CA∩C'A' — are collinear (lie on the "axis of perspectivity").
This theorem holds in projective geometry but can FAIL in the affine plane — it's equivalent to the validity of Pappus's hexagon theorem and the associativity of the coordinatizing field.