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[parent] angle of view of a line segment (Topic)

Let $ PQ$ be a line segment and $ A$ a point not belonging to $ PQ$. Let the magnitude of the angle $ PAQ$ be $ \alpha$. One says that the line segment $ PQ$ is seen from the point $ A$ in an angle of $ \alpha$; one may also speak of the angle of view of $ PQ$.

The locus of the points from which a given line segment $ PQ$ is seen in an angle of $ \alpha$ (with $ 0 < \alpha < 180^\circ$) consists of two congruent circular arcs having the line segment as the common chord and containing the circumferential angles equal to $ \alpha$.

Especially, the locus of the points from which the line segment is seen in an angle of $ 90^\circ$ is the circle having the line segment as its diameter.


\begin{pspicture}(-3,-4)(3,4) \psline[linecolor=blue](-1.73,0)(1.73,0) \rput[a](... ...lor=blue](-1.73,0)(1.73,0) \psdots[linecolor=red](-1.2,2.6)(2,1) \end{pspicture}

Note. The explementary arcs of the above mentioned two arcs form the locus of the points from which the segment $ PQ$ is seen in the angle $ 180^\circ\!-\!\alpha$.



"angle of view of a line segment" is owned by pahio.
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See Also: circumferential angle is half the corresponding central angle, Thales' theorem

Also defines:  angle of view

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Cross-references: segment, explementary arcs, diameter, circle, circumferential angles, chord, arcs, circular, congruent, locus, angle, point, line segment
There are 3 references to this entry.

This is version 9 of angle of view of a line segment, born on 2007-10-04, modified 2007-10-14.
Object id is 9980, canonical name is AngleOfViewOfALineSegment.
Accessed 881 times total.

Classification:
AMS MSC: 51F20 (Geometry :: Metric geometry :: Congruence and orthogonality)
 51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
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