V OM - I - Task 11

Given are two intersecting lines $ a $ and $ b $. Find the geometric locus of point $ M $ having the property that the distance between the orthogonal projections of point $ M $ on lines $ a $ and $ b $ is constant, equal to a given segment $ d $.
Let $A$ and $B$ denote the projections of point $M$ onto lines $a$ and $b$ intersecting at point $O$, and let $\alpha$ denote the convex angle $AOB$ (Fig. 27). The circle with diameter $OM$ passes through points $A$ and $B$, since angles $OAM$ and $OBM$ are right angles; this circle is thus the circumcircle of triangle $AOB$ and $AB = OM \sin \alpha$. If $AB = d$, then $OM = \frac{d}{\sin \alpha}$. Therefore, point $M$ lies on the circle $k$ with center $O$ and radius $\frac{d}{\sin \alpha}$.

Conversely, if a point $N$ lies on this circle $k$ (Fig. 28), then its projections $C$ and $D$ onto lines $a$ and $b$ lie on the circle with diameter $ON = \frac{d}{\sin \alpha}$; this circle is the circumcircle of triangle $COD$, so $CD = \frac{d}{\sin \alpha} \cdot \sin \measuredangle COD$. But $\measuredangle COD$ is equal to $\alpha$ or $180^\circ - \alpha$, so $\sin \measuredangle COD = \sin \alpha$ and $CD = d$.

The sought geometric locus is thus the circle drawn with the intersection of the given lines as the center and a radius equal to $\frac{d}{\sin \alpha}$, where $\alpha$ denotes one of the angles between the given lines.