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Version 8 |
| 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 $PQ$ {\em is seen from the point $A$ in an angle of $\alpha$}; one may also speak of the {\em angle of view} of $PQ$. |
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 $PQ$ {\em is seen from the point $A$ in an angle of $\alpha$}; one may also speak of the {\em angle of view} of $PQ$. |
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| 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 $\alpha$. |
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 $\alpha$. |
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| 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. |
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. |
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| \begin{center} |
\begin{center} |
| \begin{pspicture}(-3,-4)(3,4) |
\begin{pspicture}(-3,-4)(3,4) |
| \psline[linecolor=blue](-1.73,0)(1.73,0) |
\psline[linecolor=blue](-1.73,0)(1.73,0) |
| \rput[a](-2.1,-0.1){$P$} |
\rput[a](-2.1,-0.1){$P$} |
| \rput[a](2.1,-0.1){$Q$} |
\rput[a](2.1,-0.1){$Q$} |
| \psarc[linecolor=red](0,1){2}{-30}{210} |
\psarc[linecolor=red](0,1){2}{-30}{210} |
| \psarc[linecolor=red](0,-1){2}{-210}{30} |
\psarc[linecolor=red](0,-1){2}{-210}{30} |
| \psline(-1.73,0)(-1.2,2.6) |
\psline(-1.73,0)(-1.2,2.6) |
| \psline(1.73,0)(-1.2,2.6) |
\psline(1.73,0)(-1.2,2.6) |
| \psline(-1.73,0)(2,1) |
\psline(-1.73,0)(2,1) |
| \psline(1.73,0)(2,1) |
\psline(1.73,0)(2,1) |
| \rput[a](-1.08,2.25){$\alpha$} |
\rput[a](-1.08,2.25){$\alpha$} |
| \rput[a](1.72,0.75){$\alpha$} |
\rput[a](1.72,0.75){$\alpha$} |
| \psdots[linecolor=blue](-1.73,0)(1.73,0) |
\psdots[linecolor=blue](-1.73,0)(1.73,0) |
| \psdots[linecolor=red](-1.2,2.6)(2,1) |
\psdots[linecolor=red](-1.2,2.6)(2,1) |
| \end{pspicture} |
\end{pspicture} |
| \end{center} |
\end{center} |
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| \textbf{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 $180^\circ\!-\!\alpha$. |
\textbf{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 $180^\circ\!-\!\alpha$. |
