Question 2
In a plane $(P)$, two fixed lines $a,b$ and two variable lines $x,y$ are given. During their motion $x,y$ remain parallel and pass respectively through two points $A,B$ lying on “line $A$” [as printed]; $x$ meets $b$ at $C$, and $y$ meets $b$ at $D$. Through the intersection $M$ of $AB$ and $CD$, draw the line parallel to $x$, meeting $a$ at $L$ and $b$ at $N$.
(a) What can be said about $L,M,N$? Prove it and determine the locus of $M$.
(b) Let a second plane $P'$ not parallel to $P$ and a point $O$ outside both planes be given. The planes $Oa,Ob,Ox,Oy$ may intersect $P'$ or be parallel to it; when they intersect it, denote their intersections by $a',b',x',y'$ respectively. Thus there is a corresponding locus problem in $P'$ for the intersection $M'$ of “line $M$” [as printed] with $P'$. State that locus problem.
(c) In which positions of $x,y$ are $x',y'$ parallel, and in which positions are $A'D',B'C'$ parallel?
Editorial transcription note: The three quoted constructions are retained as printed. The solution explains the literal defect and separately studies a proposed correction; this is not a silently corrected official statement.
(a) What can be said about $L,M,N$? Prove it and determine the locus of $M$.
(b) Let a second plane $P'$ not parallel to $P$ and a point $O$ outside both planes be given. The planes $Oa,Ob,Ox,Oy$ may intersect $P'$ or be parallel to it; when they intersect it, denote their intersections by $a',b',x',y'$ respectively. Thus there is a corresponding locus problem in $P'$ for the intersection $M'$ of “line $M$” [as printed] with $P'$. State that locus problem.
(c) In which positions of $x,y$ are $x',y'$ parallel, and in which positions are $A'D',B'C'$ parallel?
Editorial transcription note: The three quoted constructions are retained as printed. The solution explains the literal defect and separately studies a proposed correction; this is not a silently corrected official statement.