= Solution
Two <projective triangles> are in <perspectivity from a point> $O$ when their three corresponding-vertex <projective lines> $PP'$, $QQ'$, and $RR'$ all pass through $O$.
\b[<Desargues's theorem>: <perspectivity from a point> implies <perspectivity from a line>.] More explicitly, if $PP',QQ',RR'$ are concurrent, then the three corresponding-side intersections
$$
X=PQ\cap P'Q',\qquad Y=QR\cap Q'R',\qquad Z=RP\cap R'P'
$$
are <collinear>. The usual statement assumes nondegenerate <projective triangles> and well-defined, distinct corresponding-side intersections. The <projective line> containing $X,Y,Z$ is the axis of <perspectivity from a line>. All intersections are projective: parallel lines in an <affine chart> meet on its <projective line at infinity>.
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