Given a line segment $PQ$ and a straight line $L$ not intersecting....

Question

Given a line segment $PQ$ and a straight line $L$ not intersecting the line segment. At what point on $L$ will $PQ$ subtend the greatest angle.

I am looking for a geometric result..

Thanks.

Answer 1

Hint:

Draw a family of circles through the points $P$ and $Q$ (the centers of the circles lie on perpendicular bisector of $PQ$). Among these circles there are two which are tangent to the line $L$. If the line $L$ is not perpendicular to $PQ$ one of these two circles has a smaller radius. The point of tangency of the circle and the line is the point you are looking for.

The probably simplest geometric construction of the point in question is the following (see picture):

Let $O$ be the intersection point of $PQ$ and $L$. Let $L'$ be the line through $O$ which is perpendicular to $L$. Construct points $P',Q'$ on $L'$ such that $OQ'=OQ$, $OP'=OP$, $O\in P'Q'$. Draw a circle with diameter $P'Q'$. It intersects the line $L$ in two points $X$ and $X'$. The point with acute angle $XOP$ is the point which maximizes the angle. The other point $X'$ (not shown) maximizes the angle in the other half-plane with respect to the line $PQ$.