Consider pairs of reduced fractions $m/n\lt 1$ and $p/q\lt 1$, where $m,n,p,q$ are positive integers, such that if $m/n=\tan\alpha$ and $p/q=\tan\beta$, then $\alpha+\beta=45^\circ$.
(a) Given $m,n$, determine $p,q$. When does a solution exist?
(b) Given $n,q$, determine $m,p$. When does a solution exist?
(c) Given $m,q$, determine $n,p$. When is there one solution? When are there two?
A cube $ABCD.EFGH$ has edge $AB=2a$. Consider lines $d$ meeting the three infinite lines $AE,BC,DF$ (in a special case a line may be parallel to one of these three). Let $M$ be the intersection of $d$ with $BC$, and let $m$ be the distance from $M$ to $B$.
(a) Prove that plane $MAE$ meets $FG$ at a point $E'$ at distance $m$ [the source omits the reference point], and that plane $MDF$ meets $EH$ at a point $F'$ at distance $m$ from $H$. Deduce a construction of the intersection $P$ of $d$ with face $EFGH$.
(b) Calculate the distances from $P$ to lines $EF,EH$ in terms of $m$. Find a relation between these distances independent of $m$, and hence the locus of $P$. Which part corresponds to $M$ on edge $BC$?
(c) The lines $d$ form a surface $S$. Along which lines does $S$ intersect the faces of the cube, and why?