Question 4
A fixed circle has center $O$, radius $R$, two fixed perpendicular diameters $AB,CD$, and tangent $At$ at $A$. A point $M$ moves on the circle; the lines $BM,DM$ meet $At$ at $P,Q$, respectively.
(a) Prove that the distance between $P,Q$ is the fourth proportional to the three lengths $AP,AQ$ and the diameter. Express the relation by the simplest possible formula.
(b) Construct with ruler and compass a point $M$ for which $BQ$ and $DP$ are parallel.
(c) Let $OP$ and $BQ$ meet at $N$. Guess the locus of $N$ and prove it.
(d) Let $BP,BQ$ meet the tangent at $D$ at $P',Q'$, respectively. State and prove a relation between $P',Q'$. Let $DP,DQ$ meet $BC$ at $P'',Q''$. State and prove a relation between $P'',Q''$.
Editorial clarification: use configurations where the named lines and finite intersections are defined. The unqualified source wording in (a), including its order of the three lengths, is preserved for discussion in the solution.
(a) Prove that the distance between $P,Q$ is the fourth proportional to the three lengths $AP,AQ$ and the diameter. Express the relation by the simplest possible formula.
(b) Construct with ruler and compass a point $M$ for which $BQ$ and $DP$ are parallel.
(c) Let $OP$ and $BQ$ meet at $N$. Guess the locus of $N$ and prove it.
(d) Let $BP,BQ$ meet the tangent at $D$ at $P',Q'$, respectively. State and prove a relation between $P',Q'$. Let $DP,DQ$ meet $BC$ at $P'',Q''$. State and prove a relation between $P'',Q''$.
Editorial clarification: use configurations where the named lines and finite intersections are defined. The unqualified source wording in (a), including its order of the three lengths, is preserved for discussion in the solution.