Question 171
An acute angle $\angle ABC$ and interior ray $BD$ are given, as shown. Laura is given an infinite ruler which consists of two parallel rays joined on one end by a segment perpendicular to both of them. One may place the infinite ruler onto the diagram so that one of the infinite edges (marked with an arrow) passes through any two selected points in the diagram, or so that any edge of the ruler coincides with (i.e. exactly overlaps with) a portion of a segment, ray or line already in the diagram. Once the ruler is placed, one may draw any edge of the ruler "onto the diagram", in the usual fashion. One may also plot points where any two straight objects intersect.
Using only an infinite ruler, describe how to construct points $P$ on ray $BA$ and $Q$ on ray $BC$ such that ray $BD$ intersects ray $PQ$ at a point $R$ with $PR = 2(QR)$. Then prove that your construction works.
Using only an infinite ruler, describe how to construct points $P$ on ray $BA$ and $Q$ on ray $BC$ such that ray $BD$ intersects ray $PQ$ at a point $R$ with $PR = 2(QR)$. Then prove that your construction works.